Khizar Qureshi - ir.library.dc-uoit.ca

Robust Controller Design for ActiveTrailer Steering Systems ofArticulated Vehicles Using

Multi-objective

byKhizar Qureshi

A thesis submitted to theSchool of Graduate and Postdoctoral Studies in partial

fulfillment of the requirements for the degree of

Master of Applied Science in Electrical and ComputerEngineering

Faculty of Engineering and Applied ScienceUniversity of Ontario Institute of Technology

Oshawa, Ontario, Canada

August 2019

©Khizar Ahmad Qureshi, 2019

[email protected]

Faculty Web Site URL Here (include http://www.engineering.uoit.ca/)

THESIS EXAMINATION INFORMATION

Submitted by: Khizar Ahmad Qureshi

Masters of Applied Science in Electrical and Computer

Engineering

Thesis Title: Robust Controller Design for Active Trailer Steering System of

Articulated Vehicles Using Multi-Objective Optimization

An oral defense of this thesis took place on August 6, 2019 in front of the fol-

lowing examining committee:

Examining Committee:

Chair of Examining Committee Dr. Shahryar Rahnamayan

Research Supervisor Dr. Ramiro Liscano

Research Co Supervisor Dr. Yuping He

Examining Committee Member Dr. Akramul Azim

External Examiner Dr. Jaho Seo

The above committee determined that the thesis is acceptable in form and con-

tent and that a satisfactory knowledge of the field covered by the thesis was

demonstrated by the candidate during an oral examination. A signed copy of

the Certificate of Approval is available from the School of Graduate and Post-

doctoral Studies.

STATEMENT OF CONTRIBUTIONS

Part of this work has been published in conferences as,

1. Mutaz Keldani, Khizar Qureshi, Yuping He, Ramiro Liscano. Design Op-

timization of a Robust Active Trailer Steering System for Car-Trailer Com-

binations. Society of Automotive Engineers (SAE) WCX Congress 2019

2. Mohammed Hammad, Khizar Qureshi, Yuping He. Safety and Lateral

Dynamics Improvement of a Race Car Using Active Rear Wing Control.

Society of Automotive Engineers (SAE) WCX Congress 2019.

3. Khizar Qureshi, Shahryar Rahnamayan, Yuping He, Ramiro Liscano. En-

hancing LQR Controller Using Optimized Real-time System by GDE3 and

NSGA-II Algorithms and Comparing with Conventional Method. IEEE

Congress on Evolutionary Computation 2019.

ABSTRACT

This thesis presents and evaluates an approach to the robust controller de-

sign for active trailer steering (ATS) systems to increase the safety of articu-

lated vehicles. By applying a multi-objective evolutionary algorithm (MOEA)

to the design optimization of the robust ATS controller, a series of optimal

gain values can be obtained in a single run. This allows for posteriori deci-

sion making along with flexibility to select appropriate gain for different oper-

ating conditions. The algorithm creates Pareto optimal gain values for various

speeds, thereby resulting in the robust ATS controller with an optimized gain

scheduling scheme. The research elucidates the advantages of multi-objective

algorithms over mono-objective or single-objective algorithms. For the design

optimization of the ATS controller, a benchmark investigation is conducted

to select an effective algorithm from the multi-objective algorithms, including

GDE3, NSGA-II, NSGA-III, SPEA2 and MOPSO. A modular framework is intro-

duced for co-simulations conducted in the CarSim-Simulink/Matlab environ-

ment, with which the vehicle and controller parameters can be optimized. The

method ensures that a robust ATS controller with optimized feedback control

gains, as well as satisfaction of design criteria and constraints. This research

proposes a framework to generate a multi-dimensional look-up table using the

multi-objective evolutionary algorithm for a general dynamic system controlled

by a feedback controller. The optimized look-up system can be used to improve

the robustness of control systems in real-world applications.

AUTHORS DECLARATION

I hereby declare that this thesis consists of original work of which I have au-

thored. This is a true copy of the thesis, including any required final revisions,

as accepted by my examiners.

I authorize the University of Ontario Institute of Technology to lend this thesis

to other institutions or individuals for the purpose of scholarly research. I fur-

ther authorize University of Ontario Institute of Technology to reproduce this

thesis by photocopying or by other means, in total or in part, at the request of

other institutions or individuals for the purpose of scholarly research. I under-

stand that my thesis will be made electronically available to the public.

Khizar Ahmad Qureshi

iv

ACKNOWLEDGEMENTS

I want to extend my utmost gratitude to Dr. Ramiro Liscano and Dr. Yuping

He, for giving me the opportunity to pursue my masters at UOIT. You ensured

I was on the right track with my research and provided me with continued

financial support to guarantee I could concentrate completely on my studies.

It has been an absolute privilege to work with you both. In the ever changing

path of life, the constant, for me, will be my appreciation and remembrance of

you two. Thank You.

I want to make sure I thank my wonderful wife, Ayesha, who supported me

through my education. She always believed in me and created a wonderful

environment at home to ensure success. She put her life on hold to push me

forward, and for that I would be eternally grateful. My life would be incomplete

without you.

v

Contents

Thesis Examination Information i

Statement of Contributions ii

Abstract iii

AUTHORS DECLARATION iv

Acknowledgements v

List of Figures ix

List of Tables xiii

Abbreviations xiv

1 Introduction 11.1 Problem with Conventional Tuning Methods . . . . . . . . . . . 2

1.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Thesis Contributions and Novelty . . . . . . . . . . . . . . . . . . 5

1.3.1 MOEA for ATS Controllers . . . . . . . . . . . . . . . . . . 51.3.2 MOEA Optimized Gain Scheduled ATS Controllers . . . 61.3.3 Modular Approach for Design Optimization of Control

Systems Using CarSim-MATLAB/Simulink Co-Simulations 61.3.4 Adaptive Driver Model with Varying Reaction Time and

Vehicle Forward Speed . . . . . . . . . . . . . . . . . . . . 71.3.5 Performance Analysis of MOEAs for Control System Tuning 7

1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Literature Review 102.1 Evolutionary Algorithms Applied to Control Systems . . . . . . . 102.2 Linear Quadratic Regulator Technique . . . . . . . . . . . . . . . 11

vi

Contents vii

2.2.1 Design Optimization of LQR Controller . . . . . . . . . . 122.2.2 Difference in Approach . . . . . . . . . . . . . . . . . . . . 13

2.3 Unstable Modes of Car-Trailer Combinations . . . . . . . . . . . . 142.4 Stabilizing Articulated Vehicles . . . . . . . . . . . . . . . . . . . 15

2.4.1 Passive Steering System . . . . . . . . . . . . . . . . . . . . 162.4.2 ATS Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Multi-Mode Steering Control . . . . . . . . . . . . . . . . 172.4.4 Differential Braking . . . . . . . . . . . . . . . . . . . . . . 172.4.5 Teslas Approach to Jackknifing Prevention . . . . . . . . . 18

2.5 Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Optimal Pareto-front . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Knee-point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8 Non-dominated Sorting Genetic Algorithm . . . . . . . . . . . . 212.9 Generalized Differential Evolution . . . . . . . . . . . . . . . . . 23

2.9.1 GDE3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . 24

3 Testing the Base Algorithm: Differential Evolution 263.1 PID Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Tuning methods for PID controllers . . . . . . . . . . . . . . . . . 283.3 DE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Island distribution of DE . . . . . . . . . . . . . . . . . . . 313.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Steady State Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 Second Order System . . . . . . . . . . . . . . . . . . . . . . . . . 353.7 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . 35

3.7.1 Results and Analysis . . . . . . . . . . . . . . . . . . . . . 363.8 Convergence Speed Comparison . . . . . . . . . . . . . . . . . . . 39

3.8.1 Results and Analysis . . . . . . . . . . . . . . . . . . . . . 403.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 GDE3 and NSGA-II Comparison on LQR Based Aircraft Pitch Control 424.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Multiple Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Ensuring Stability and Controllability . . . . . . . . . . . . . . . 474.5 Optimization Parameters and Pre-compensation . . . . . . . . . 484.6 Complexity of Case Study . . . . . . . . . . . . . . . . . . . . . . 494.7 LQR Controller for Aircraft Pitch Control . . . . . . . . . . . . . 49

4.7.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . 514.7.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Design Optimization of LQR-based ATS Controller Using GDE3 565.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Rearward Amplification (RWA) . . . . . . . . . . . . . . . . . . . 57

Contents viii

5.3 Path-Following Off-Tracking . . . . . . . . . . . . . . . . . . . . . 595.4 Design ATS Controller . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4.1 3-DOF Yaw-plane Car-trailer Model . . . . . . . . . . . . 605.5 Single-lane Change Testing Maneuver . . . . . . . . . . . . . . . 635.6 Testing Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 645.7 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 65

5.7.1 Critical Speed Analysis . . . . . . . . . . . . . . . . . . . . 655.8 Forward Speed Variation . . . . . . . . . . . . . . . . . . . . . . . 665.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 Design Optimization of Gain Scheduling Controllers Using GDE3 andClosed-loop Co-Simulation 726.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 CarSim Model and Methodology . . . . . . . . . . . . . . . . . . . 74

6.2.1 Built-in Driver Model Offered from CarSim . . . . . . . . 756.2.2 Evaluation Criteria . . . . . . . . . . . . . . . . . . . . . . 77

6.3 Gain Scheduling Controller . . . . . . . . . . . . . . . . . . . . . 776.4 Modular Design Methodology . . . . . . . . . . . . . . . . . . . . 796.5 Overcoming System Limitation with Innovization . . . . . . . . . 816.6 Algorithm Comparison . . . . . . . . . . . . . . . . . . . . . . . . 836.7 Results: GDE3 Generated GSC vs Passive . . . . . . . . . . . . . . 856.8 Results: Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . . 926.9 Important Observation . . . . . . . . . . . . . . . . . . . . . . . . 96

7 Conclusions 987.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.2 Recommendations for Future Studies . . . . . . . . . . . . . . . . 99

Bibliography 101

List of Figures

2.1 Point B is the knee-point. This image is taken from [51] . . . . . 22

3.1 Responses for DE1 and MATLAB Tuner and Random Value . . . 383.2 Responses for DE2 and MATLAB Tuner and Random Value . . . 39

4.1 Feedback system model with LQR and decision support system.Open-loop plant is the aircraft pitch model. Controller gains areobtained by the algorithm, which can be dynamically changedby the decision system based on environmental factors. . . . . . 46

4.2 Feedback system model for aircraft pitch control with GDE3 . . 514.3 Pareto-front obtained by GDE3 and NSGA-II on aircraft pitch

control model. (Note: The designer can choose any LQR gainmatrix value from the Pareto-front, based on ST and RT require-ments.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Response of the aircraft pitch control system with ST prioritized. 524.5 Response of the aircraft pitch control system with RT prioritized. 524.6 Response of the aircraft pitch control system with knee-point. . 534.7 Response of the aircraft pitch control system obtained from [66]. 53

5.1 Schematic representation of the 3-DOF yaw-plane car-trailer model 615.2 A representation of an SLC steering input . . . . . . . . . . . . . 645.3 Lateral acceleration of the car at the critical speed 79.2km/h (with-

out ATS controller) . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Lateral acceleration of the trailer at the critical speed 79.2km/h

(without ATS controller) . . . . . . . . . . . . . . . . . . . . . . . 665.5 Optimal Pareto-front at 80km/h . . . . . . . . . . . . . . . . . . . 675.6 Optimal Pareto-front at 90km/h . . . . . . . . . . . . . . . . . . . 675.7 Optimal Pareto-front at 100km/h . . . . . . . . . . . . . . . . . . 675.8 Optimal Pareto-front at 110km/h . . . . . . . . . . . . . . . . . . 675.9 Lateral acceleration of the car at 80km/h . . . . . . . . . . . . . . 685.10 Lateral acceleration of the trailer at 80km/h . . . . . . . . . . . . 685.11 Lateral acceleration of the car at 90km/h . . . . . . . . . . . . . . 685.12 Lateral acceleration of the trailer at 90km/h . . . . . . . . . . . . 685.13 Lateral acceleration of the car at 100km/h . . . . . . . . . . . . . 685.14 Lateral acceleration of the trailer at 100km/h . . . . . . . . . . . 685.15 Lateral acceleration of the car at 110km/h . . . . . . . . . . . . . 695.16 Lateral acceleration of the trailer at 110km/h . . . . . . . . . . . 69

ix

List of Figures x

6.1 Car sub-model developed in CarSim . . . . . . . . . . . . . . . . 746.2 Trailer sub-model developed in CarSim . . . . . . . . . . . . . . . 756.3 Car-trailer model with the built-in driver model. . . . . . . . . . 756.4 Predefined trajectory for the closed-loop single lane-change test-

ing maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.5 System model using modular blocks. State Information from top

to bottom: Yaw-rate of car, Yaw-rate of trailer, lateral speed of carand lateral speed of trailer. . . . . . . . . . . . . . . . . . . . . . . 80

6.6 Algorithm comparison based on two design PFOT. . . . . . . . . 846.7 Trajectory of Car at 120km/h for Algorithm Comparison . . . . . 856.8 Trajectory of Trailer at 120km/h for Algorithm Comparison . . . 856.9 Lateral Acceleration of Car at 120km/h for Algorithm Comparison 856.10 Lateral Acceleration of Trailer at 120km/h for Algorithm Com-

parison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.11 Optimal Pareto-front for 80 km/h and 0(s) driver model reaction

time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.12 Optimal Pareto-front for 90 km/h and 0(s) driver model reaction

time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.13 Optimal Pareto-front for 100 km/h and 0(s) driver model reac-

tion time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.14 Optimal Pareto-front for 110 km/h and 0(s) driver model reac-

tion time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.15 Optimal Pareto-front for 120 km/h and 0(s) driver model reac-

tion time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.16 Optimal Pareto-front for 80 km/h and 0.1(s) driver model reac-





tion time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.21 Trajectory of the car at 80 km/h and 0(s) driver model reaction

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.22 Trajectory of the trailer at 80 km/h and 0(s) driver model reac-

tion time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.23 Trajectory of the car at 90 km/h and 0(s) driver model reaction

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.24 Trajectory of the trailer at 90 km/h and 0(s) driver model reac-

tion time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.25 Trajectory of the car at 100 km/h and 0(s) driver model reaction

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

List of Figures xi

6.26 Trajectory of the trailer at 100 km/h and 0(s) driver model reac-tion time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.27 Trajectory of the car at 110 km/h and 0(s) driver model reactiontime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89


6.29 Trajectory of the car at 120 km/h and 0(s) driver model reactiontime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89


6.31 Trajectory of the car at 80 km/h and 0.1(s) driver model reactiontime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.32 Trajectory of the trailer at 80 km/h and 0.1(s) driver model reac-tion time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


6.34 Trajectory of the trailer at 90 km/h and 0.1(s) driver model reac-tion time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


6.36 Trajectory of the trailer at 100 km/h and 0.1(s) driver model re-action time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90





6.41 GSC Trajectory of the car at 80 km/h and 0.0(s) driver modelreaction time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.42 GSC Trajectory of the trailer at 80 km/h and 0.0(s) driver modelreaction time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92






List of Figures xii












List of Tables

3.1 The table lists the assigned labels of the variants of DE based onmutation schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 The experimental results for DE1 to DE6 and the average RT, STand OS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 The experimental results for DE7 to DE12 and the average RT,ST and OS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 MATLAB PID tuner and random value results . . . . . . . . . . . 373.5 Rank of DE Variants in Convergence Comparison . . . . . . . . . 40

4.1 Control parameters of Optimization . . . . . . . . . . . . . . . . 484.2 Comparison Aircraft Pitch Control . . . . . . . . . . . . . . . . . 54

5.1 Car-trailer combination parameters for ATS controller design . . 635.2 RWA Result Comparison. . . . . . . . . . . . . . . . . . . . . . . 695.3 Peak Lateral Acceleration (g) of Trailer Result Comparison . . . . 69

6.1 Vehicle parameter values. . . . . . . . . . . . . . . . . . . . . . . . 746.2 Possibilities of GSC Look-up Table . . . . . . . . . . . . . . . . . 796.3 RWA Comparison GS vs W/O GS . . . . . . . . . . . . . . . . . . 95

xiii

Abbreviations

EA Evolutionary Algorithm

NSGA Non-dominated Sorting Genetic Algorithm

GA Genetic Algorithm

DE Differential Evolution

PSO Particle Swarm Optimization

GDE Generalized Differential Evolution

AHV Articulated Heavy Vehicles

RWA Rearward Amplification

LQR Linear Quadratic Regulator

DM Decision Maker

MOEA Multi Objective Evolutionary Algorithm

PAES Pareto-archived Evolutionary Strategy

SPEA Strength Pareto Evolutionary Algorithm

MOGA Multi Objective Genetic Algorithm

ATS Active Trailer Steering

xiv

Chapter 1

Introduction

Control systems are essential in the modern world to tackle ever-changing en-

vironmental disturbances while meeting adequate safety standards. Control

systems must be robust and appropriately tuned for each application to en-

sure reliability and safety. Evolutionary algorithms are proven to be effective

in offline tuning of control systems for various applications [1]. Depending on

the complexity of a problem, evolutionary algorithms can even take days to

compute an optimal result. This approach is logical for systems with no en-

vironmental dependencies. If a system is subject to external disturbances, its

behaviour changes and an optimal controller may not be able to stabilize the

system, resulting in a failure. To tackle this problem, this thesis proposes an

approach to use evolutionary multi-objective optimization, for offline tuning

of a control system at multiple dependencies, creating a comprehensive opti-

mized look-up table to increase robustness of the system. This method will also

provide posteriori decision making capabilities for multiple objectives.

1

Chapter 1 2

1.1 Problem with Conventional Tuning Methods

The need for complicated and robust control algorithms is ever increasing for

real-time systems. Ranging from autonomous cars to fully-automated intelli-

gent manufacturing, all require complex controllers to increase efficiency, safety

and reduce costs. The essence of a control system is to measure the output to

provide corrective feedback to ensure that the system operates with the desired

performance even though many environmental changes occur. With the shift

towards Internet of Things and smart cities, the need for effective control meth-

ods gets very demanding. Optimal control techniques have a high potential to

be enhanced by optimization. Some conventional strategies, to tune optimiza-

tion techniques, are used in [2]. The obtained values are mainly system specific

and tailored by adjusting the weights of gain matrices for the control method.

These methods often require a deep understanding of the model and in-depth

knowledge of the technique and the system, which is under enhancement. The

thesis deviates from conventional techniques, and employs population-based

meta-heuristics to tune weighting matrices to achieve optimal solutions.

1.1.1 Motivation

Car-Trailer combinations are a type of Articulated Vehicles (AV), which is a

combination of multiple vehicle units [3]. An AV can carry a greater payload,

compared to a single unit vehicle, due to an additional trailer while using the

same engine [3]. This allows for a reduction in pollution and increases cost-

effectiveness for customers. The trailers, on an AV, are joined with a mechanical

coupling, called hitch [4]. Because of this coupling, some new problems arise.

Chapter 1 3

The first challenge with an AV is the size of the vehicle, an AV is much larger

than a normal car. Any collision or roll-over of an AV can result in damage not

only to the AV but neighboring/surrounding vehicles and pedestrians. Thus,

ensuring the safety of an AV is paramount but rather difficult.

The second challenge, with car-trailer combinations, is the nonlinear dynamic

behaviors. The nonlinear dynamics of car-trailers may lead to severe traffic acci-

dents, such as trailer sway, jackknifing, and rollover. A car-trailer combination

can be linearized by assumptions, for example, fixing the forward speed to be a

constant and the steering angle to be small [5].

The first challenge is tackled by using an ATS system and optimizing it us-

ing multi-objective algorithms. The second challenge is addressed by using a

multi-objective optimized gain scheduling controller, with a two-dimensional

look-up table. The similar problems with car-trailer combinations also plague

the trucking industry. The amazon e-Commerce effect [6], discussed by Gwen

Mortiz, has greatly impacted the trucking industry. Users can purchase goods

online and get them shipped to their houses, workplaces, PO Boxes etc. These

products have increased the already burdened trucking industry, which now

must make several trips daily, for express shipping services required by con-

sumers. Increasing the number of trucks to carry payload increases emissions,

requires more drivers and have more cost of operation. Transport trucks are

also involved in 20% of road crashes in Ontario [7]. The way to increase the

safety of transport vehicles is to apply advanced optimized control systems,

which is the motivation behind this thesis. Within the limitation of time and

resources, this thesis only focuses on the advanced optimized control systems

Chapter 1 4

for car-trailer combinations. The proposed methodology can be extended to ar-

ticulated heavy vehicles (AHVs), since both car-trailer combinations and AHVs

share similar lateral dynamics.

1.2 Problem Statement

There are no perfect, noise and uncertainty free environments for a vehicle sys-

tems to operate. An optimal control system provides a way to tackle the ever-

changing environments of real-time systems. In the design and modeling stage,

the systems are functioning using a set of optimal parameter values. The design

expectations have to be constrained to allow for that single set of parameter val-

ues to meet all requirements. The tuning methods (state space exploration) [2]

are not fast enough to be implemented in safety-critical real-time systems. This

is why designers have to make a tradeoff. Each solution also requires compre-

hensive testing to ensure that this solution results in a stable system, to avoid

any catastrophic failure. In this thesis, an approach is proposed to tackle both

of these problems. This method produces a set of solutions for multiple objec-

tives without doing weighted aggregation for multiple objectives. The designer

only needs to choose one of those solutions, depending on the operating con-

dition of a real-time system, instead of re-running the optimizer and finding a

new solution. This approach allows for more flexibility in designing controllers

for real-time systems. These solutions are calculated for varying environmen-

tal factors to create multi-dimensional look-up tables, which is used with a gain

scheduling controller.

Chapter 1 5

1.3 Thesis Contributions and Novelty

Design of a control system is usually based on more than one objective. A sim-

ple example can be rise-time and settling-time of an inverted pendulum [8].

When evolutionary algorithms are used, all objectives are aggregated into a sin-

gle fitness function or a single objective function [9]. The aggregated objective

function is then evaluated to ascertain the results.

For two objectives, if a MOEA is used, a set of solutions is obtained where each

solution has two fitness value, one for each objective. The optimal set of such

values is called the optimal Pareto-front. This approach has not been utilized in

the design of ATS systems for car-trailer combinations. The following subsec-

tions introduce the thesis contributions, which are based on the control system

tuning for ATS systems of car-trailer combinations.

1.3.1 MOEA for ATS Controllers

Evolutionary algorithms have been used to tune ATS systems [10, 11]. Mono-

objective algorithms are used for the tuning of the ATS controllers, e.g. ge-

netic algorithms. Using mono-objective algorithm eliminates any possibility of

choice, this is further discussed in section 4.3.

The first contribution of the thesis is to tune an ATS controller for car-trailer

combination, and to provide the system designer with a choice from an optimal

set of solutions. Ensuring each solution is stable and applicable within design

goals. This is done by using MOEAs. The MOEA studied in this thesis is known

as generalized differential evolution (GDE3), which is outlined in section 2.9.

GDE3 is used for the design of ATS controllers in Chapters 5 and 6.

Chapter 1 6

1.3.2 MOEA Optimized Gain Scheduled ATS Controllers

The car-trailer combination is a non-linear system. It is linearized using as-

sumptions to generate a state space representation, which is then used in con-

trollers design, e.g., linear quadratic regulator (LQR), to develop a feedback

control system. This is the approach used in [11]. The second contribution of

the thesis is the provision of an ATS system using an optimized gain scheduling

controller (GSC) for car-trailer combinations. The LQR-based ATS controller is

tuned using MOEA at various speeds to provide a lookup table for the system

designer to implement in the control strategy. The GSC makes the system more

robust. This is discussed in detail in Chapter 6.3.

1.3.3 Modular Approach for Design Optimization of Control

Systems Using CarSim-MATLAB/Simulink Co-Simulations

Control system tuning requires system-specific knowledge, especially if a lin-

earized model is to be generated. An example is the 3 degrees of freedom (DOF)

car-trailer model that is derived in Chapter 5. The steps involve: 1) generating

a linearized model to represent the non-linear system by using certain assump-

tions, and 2) testing to ensure that the linearized model derived captures the

essential dynamic features if the non-linear system. This process is cumber-

some and requires knowledge of the mechanical system to a great extent. In

Chapter 6 of this thesis, an approach is used, which eliminates the use of linear

control strategies and replace it with an evolutionary strategy, which can work

without the requisite of a set of linear equations of the system.

Chapter 1 7

1.3.4 Adaptive Driver Model with Varying Reaction Time and

Vehicle Forward Speed

The thesis explores two parameters for gain scheduling for the built-in driver

model offered in CarSim software: 1) driver model reaction time, and 2) vehicle

forward speed. The built-in driver model in CarSim can be used to drive the

virtual car-trailer combination to follow a prescribed single lane-change (SLC)

trajectory. GDE3 is used to create a gain scheduling scheme, which considers

the reaction time of the driver model and the varying vehicle forward speed.

This allows for adjusting not only the control gain matrix of ATS controller, but

also the essential parameter of the driver model.

1.3.5 Performance Analysis of MOEAs for Control System Tun-

ing

The MOEA has not been used for the design optimization of ATS system for car-

trailer combinations. The last contribution of this thesis is to compare popular

MOEAs by means of evaluating the respective ATS controllers designed car-

trailer combinations.

1.4 Thesis Organization

The rest of the thesis is organized as follows. Chapter 2 provides a literature

review of control system tuning using evolutionary algorithms, problems with

car-trailer combinations and how they can be solved. In Chapter 2 the building

Chapter 1 8

blocks of the thesis are also discussed, which include the car-trailer combina-

tion, optimization algorithms, multi objective optimization techniques and the

linear quadratic regulator (LQR) control technique. The verification of opti-

mization search algorithms starts from Chapter 3 where a PID controller is

tuned using differential evolution to test which variant is the best, this is later

used in Chapters 5 and Chapter 6. Chapter 4 illustrates a comprehensive case

study to compare genetic algorithm based MOEA against a differential evolu-

tion based MOEA. Confirming that the algorithm performing well on existing

models is essential before generating the models and testing the algorithm.

In Chapter 5, a 3-DOF car-trailer model is generated using a multi-body dy-

namic software. Then, the linear 3-DOF car-trailer model is used to design the

ATS controller using the LQR technique. The LQR-based ATS controller is op-

timized based on open-loop dynamic simulation using the algorithm discussed

in 4. Lastly, Chapter 6 is the crux of all previous chapters. It applies every-

thing tested before, with a co-simulation with a Car-Trailer model from CarSim

mechanical simulator. It reaffirms the results and provides insights into gain

scheduling for non-linear systems, tuning based on adaptive driver model with

varying reaction time and vehicle forward speed and algorithm comparison.

Chapter 6 also provides recommendations for future studies.

This thesis involves five major steps to conclude. The first step is to decide the

base evolutionary algorithm (EA) from a large selection of database of EA and

their variants. Chapter 3 focuses on comparing different variants of differential

evolution (DE). The results from Chapter 3 provide the mutation scheme and

algorithm which is used as the base algorithm in Chapters 4 to 6. Generalized

Chapter 1 9

Differential Evolution (GDE3) is the corresponding multi-objective evolution-

ary algorithm (MOEA) of differential evolution. In Chapter 4 the results re-

ceived from Chapter 3 are used with a multi-objective scheme and compared

to a popular algorithm. The GDE3 variant tested and verified in Chapter 4 is

then used in Chapters 5 and 6. In Chapter 5, an LQR controller is tuned using

GDE3 variant from the previous chapter and applied to a 3-DOF car-trailer ar-

ticulated vehicle. The controller, objectives and constraint design from Chapter

5 is used in Chapter 6, which concludes the thesis results.

Chapter 2

Literature Review

The literature review focuses on methods for tuning control systems and sta-

bility of articulated vehicles while providing some background information on

general techniques used in this thesis.

2.1 Evolutionary Algorithms Applied to Control Sys-

tems

Evolutionary algorithms (EAs) and MOEAs are proven to be effective in offline

tuning of control systems for various applications [1]. The authors in [12] stud-

ied the effectiveness genetic algorithm (GA) in tuning of PI and LQR controllers

for boiler-turbine plant.

Non-dominated sorting genetic algorithm (NSGA-II) was used as MOEA to op-

timize HVAC control system [13]. NSGA- II achieves an improvement, within

the design constraints, while considering multiple objectives [13].

A MOGA was used to generate an optimal Pareto-front, allowing for multi-

criterion decision making, in an energy conservation application for building

10

Chapter 2 11

design [14].

As discussed by the authors of [15], EAs were employed in most of the LQR im-

plementations as optimizer [16, 17]. The authors of [17] concluded that differ-

ential evolution, along with other continuous optimizers, outperforms genetic

algorithms, which reinforces the selection of Differential Evolution as the core

optimization method. All the above studies, collectively, show that the LQR

control method is used in real-time systems and that evolutionary algorithms

achieve a better solution than other conventional methods or even fuzzy logic

controllers.

An ATS control system, using the LQR-based controller, for articulated vehicles

is designed and optimized by GA in [18] and shows that it is superior to other

studies. A GA optimized active trailer differential braking system is outlined

in [19] for a car-trailer combination. The aforementioned studies along with

engineering applications show that evolutionary search algorithms are ideal

for tuning control systems in a wide array of applications [1].

2.2 Linear Quadratic Regulator Technique

The LQR technique was introduced by Kalman [20, 21]. The LQR controller is

a feedback controller used to provide optimal control for a dynamic system to

ensure operation at minimum cost. A continuous-time linear system is defined

as follows:

x = Ax +Bu (2.1)

Where A represents the states of the system which is a square matrix, B is the

input matrix, x is the state vector, and u is the control vector. For the optimal

Chapter 2 12

controller design, the continuous-time LQR cost function is defined as

J =ˆ ∞

0(xTQx + utRu + 2xTNu)dt (2.2)

Where Q and R matrices are used to find an optimal gain matrix. J is the in-

ternal cost function of the LQR control method, whereas the optimizer will try

to minimize objectives, Settling-time (ST) and Rise-time (RT), by changing Q

and R matrices, which in turn change how the LQR cost function performs in a

closed-loop.

The optimizer manipulates the Q and R matrices to minimize the objective

function described in Eq.(2.2). These matrices, which vary in size from system

to system, consist of the variables and their number corresponding to the di-

mension, which determines the complexity of the system. The LQR controller

gives a better overall performance when compared to a PID controller, but the

values are harder to attain [22]. In a PID controller, there are three control

gains, i.e. Kp , Ki and Kd . In terms of complexity, the dimension of the PID

problem is 3. For an aircraft pitch control, the dimension of the problem is 9,

as the Q matrix is a 3 ∗ 3 matrix. In [22] authors demonstrate that optimization

of the control matrix of LQR is harder than a PID controller.

2.2.1 Design Optimization of LQR Controller

The research in this thesis combines two areas: utilizing the LQR technique

for ATS controller design, and using evolutionary algorithms to optimize the

LQR control parameters. The LQR technique has been utilized to design the

Chapter 2 13

ATS controller for an AHV [23]. The study used a genetic algorithm to opti-

mize a LQR controller for ATS of AHVs, and demonstrates the strength of the

LQR controller, and also elaborates on the importance of optimization [23]. The

work in [24] uses an LQR controller to control an anti-roll bar, which prevents

vehicle rollover and they demonstrate how the LQR controller can provide sta-

bility to vehicular systems.

The LQR controller can be used to mitigate the risks in many safety-critical

systems, for instance ATS Systems to reduce the risk of rollover and jackknifing

in AHV [3, 25, 26]. It also has been used in Active Suspension System [27] to

provide a smoother and safer driving. This work also shows that a LQR con-

troller performed better than comparative controllers, in active suspension, and

reached the performance of industry implemented suspension systems in some

situations. A comparative analysis of multiple control strategies for active steer-

ing system has been conducted and found that the LQR outperforms fuzzy logic

controllers (FLC) and improves low-speed maneuverability and high-speed sta-

bility [28]. An LQR controller provides a faster and stable response, in real-

time magnetic levitation performance, as compared to FLC and PID controllers

[29, 30]. It is presented in [15] that an LQR controller has been successfully

used to control aircraft pitch.

2.2.2 Difference in Approach

Tuning an LQR controller is a big part of this thesis, as the LQR controller is

used in ATS system for the car-trailer combination to ensure stability. Similarly

an LQR controller is designed for the control system for aircraft pitch control.

Chapter 2 14

The difference in approach to both applications is the use of MOEA, GDE3 for

tuning. GDE3 is proven to be better than the counterpart genetic algorithms.

2.3 Unstable Modes of Car-Trailer Combinations

A typical car-trailer combination consists of a leading vehicle unit and a trail-

ing unit, which are connected with a mechanical hitch [4]. The hitch connection

between the vehicle units generates several mechanical constraints, making the

dynamics and kinematics of this combination more complex to investigate com-

pared to a single-unit vehicle, e.g., car or truck. Due to the unique dynamics

of the combination and the mechanical constraints, the car-trailer combination

usually faces unstable states, which could lead to fatal accidents. Typical un-

stable motion modes lead to crash of car-trailer combinations are trailer-sway,

jackknifing and rollover [31]. Several reasons could lead to each of the afore-

mentioned unstable motion modes, e.g., high-speed evasive maneuvers, cross-

winds, road conditions, and payload variation of the trailer [32].

Trailer sway is a dynamic phenomenon where the trailer exhibits a fishtailing

angular motion around the hitch. The low yaw-damping ratio of the trailer

is the primary cause of the trailer sway. The forward speed of the combina-

tion directly influences this damping ratio; when traveling on a straight line at

a critical speed, the corresponding yaw damping ratio decreases to zero [33];

while traveling above the critical speed, and the car-trailer combination loses

its lateral stability with increasing amplitude of angular oscillation [34]. The

damping ratio is an important criterion for determining the yaw-stability of car-

trailer combinations. If the combination is exposed to external disturbances,

Chapter 2 15

e.g., crosswind, it may cause the trailer to lose traction, resulting in swinging

and skidding of the combination [32].

The second is Roll-over, as the name suggests, this is when the rear trailer, due

to undue forces [35, 36] and shift in center of gravity, might roll-over, causing

the entire vehicle to rollover on the road.

The third unstable motion mode is known as Jackknifing. It is heavily depen-

dent on the relative braking force distribution between the car and the trailer.

In the event of panic breaking, if the articulated angle reached a certain limit, it

could result in jackknifing where the driver has no control over it [37]. Jackknif-

ing usually occurs during curve negotiation with heavy braking and external

disturbances, for example crosswind [38].

2.4 Stabilizing Articulated Vehicles

The detrimental effect of the three aforementioned unstable motion modes is

reduced by using various strategies. These strategies can be passive and active.

Passive strategies require no additional energy or complicated implementation

whereas active strategies do require additional energy to provide stabilization.

The strategy considered in this thesis is the ATS system, which is an active

control strategy. This system requires a steering actuator to steer the wheels of

the trailer of a car-trailer combination. The distinguished feature of the thesiss

approach is the use of GDE3 for tuning the LQR controller, instead of the mono-

objective based method for tuning the controller. Other control strategies are

reviewed in this section.

Chapter 2 16

2.4.1 Passive Steering System

Passive steering systems (PSSs) have been used to improve low-speed maneu-

verability. There are various types of PSSs: Self-Steering, Command Steering

and Pivotal Bogie Systems, etc. A comparison of these is given in [39]. The

common element among all the systems is how they deal with maneuverability.

Curved path turning, e.g., 90-degree turning, frequently occur at low speeds.

During a turn, the trailer is required to follow the trajectory of the car. When

the car turns, a PSS uses geometric the relationship between front axles and the

trailer axles to ensure the trailer to follow the same path of the car [39, 40]. A

PSS may promote a greater tail-swing. Tail-swing is a dynamic phenomenon

of a car-trailer combination, where the trailer oscillates angularly around the

hitch at the rear end of the car. At high speeds, the lateral stability is crucial

and should be enhanced, while PSSs performs poorly in terms of high-speed

lateral stability [41]. For articulated vehicles, there exists a distinguished dy-

namic phenomenon, called rearward amplification (RWA), implying that the

trailer shows larger lateral motion than that of the car. Generally, to ensure the

high-speed lateral stability of an articulated vehicle, the PSS should be locked

or disabled above a high speed, e.g., 70 km/h.

2.4.2 ATS Systems

An ATS system may allow an articulated vehicle to operate with a fix steer ratio

between the leading vehicle unit steering angle in relation to the trailer axle

steering angle [42]. Using an ATS, the trade-off between the low-speed ma-

neuverability and the high-speed lateral stability may be mitigated. An ATS

system may improve the RWA measure at high speeds and the path-following

Chapter 2 17

off-tracking (PFOT) measure at lower speeds, without having to use multiple

modes of operation and worrying about switching between them. An ATS

system is intended to manipulate the trajectory of the trailer and the forces/-

torques generated between trailer tires and the road surface, thereby leading to

improving the trailers path-following capability and enhancing the lateral sta-

bility. To implement an ATS system, a controller should be designed. The LQR

control technique has been extensively used for the design of ATS controllers.

To achieve desired performance, the control parameters of the LQR-based ATS

controller may be optimized.

2.4.3 Multi-Mode Steering Control

As discussed earlier, reducing PFOT increases RWA, and vice versa. Using both

active and passive steering systems together is one of the acceptable solutions

to the problem [42, 43]. Depending on the operation of the vehicle, the steering

system switches between Active and Passive Steering. This in turn grants ma-

neuverability at lower speeds and stability at higher speeds. The disadvantage

of the multi-mode system is the great increase in steering system complexity.

Implementing a dual-steering system on existing trucks/AHVs is very difficult

and costly.

2.4.4 Differential Braking

Fancher discussed the use of differential braking for improving the lateral sta-

bility of AHVs. Differential breaking helps reduce RWA [44]. In [44] the results

showed that using their differential breaking model, RWA was reduced to 1.7

from 2.3 at high speeds. This is achieved by applying different pressure at the

Chapter 2 18

braking actuators of the right and left wheels on an axle when the differen-

tial braking system is actuated. Although the ideal RWA value of 1.0 was not

achieved, a great reduction was still achieved. The downside of using differ-

ential braking is the toll on the braking system. The brakes are more prone

to heating up. Hence, this system must be managed carefully where it is only

turned on when necessary

2.4.5 Teslas Approach to Jackknifing Prevention

Tesla unveiled their electric semi-trucks in 2017 [45]. One of the important

safety features they unveiled was Jackknifing prevention. The approach by

Tesla is to use electric motors on individual tires and a dynamic control sys-

tem. The control system can get feedback from each motor and various sensors

around the vehicle. Whenever a danger of Jackknifing occurs, the motors ac-

tivate to counteract the action, thus preventing jackknifing. This approach is

expensive and impractical for current powertrains. None of these methods, in-

dividually, solve the entire problem, under all operating conditions. The most

promising method to use is the ATS technology. The research for this thesis

leads to the opinion that an ATS system may be optimized to provide an ac-

ceptable RWA and PFOT values, when used with an LQR controller.

2.5 Differential Evolution

Storn and Price introduced Differential Evolution (DE) [46] in 1995. It is a

population-based metaheuristic (P-metaheuristic) evolutionary algorithm (EA),

which has been inspired by a genetic algorithm (GA). DE consists of mutation,

Chapter 2 19

crossover, and selection; the algorithm performs best for continuous-valued

problems. The working principle of DE is alike GA. It starts with popula-

tion initialization. During initialization all members are assigned fitness val-

ues according to fitness functions, these values correspond to how good these

population members are for solving the problem. Following initialization, the

selection is made at random from the population space, anywhere between 2

to 4 members are chosen, depending on the utilized mutation scheme. There

are more than ten mutation variants in DE [47]. The fitness of the population

member, once chosen, is evaluated after it undergoes mutation and recombi-

nation. After applying the crossover and calculating the resulted offspring’s

fitness value, the best parent and offspring is selected to the next generation,

in fact, the selection follows a greedy strategy. DE has a specific set of control

parameters: Population Size (Np), Crossover rate (Cr), mutation scaling fac-

tors, and mutation schemes [47]. Seven different mutation schemes are listed

as follows,

V = X1 +F ∗ (X2 −X3) (2.3)

V = X1 +F ∗ (X2 −X3 +X4 −X5) (2.4)

V = Xbest +F ∗ (X1 −X2 +X3 −X4) (2.5)

V = X3 +G ∗ (Xbest −X3) +G ∗ (X1 −X2) (2.6)

V = Xbest +F ∗ (X1 −X2) (2.7)

V = Xi +G ∗ (X3 −Xi ) +F ∗ (X1 −X2) (2.8)

Chapter 2 20

V = Xi +G ∗ (Xbest −Xi ) +G ∗ (Xi −X2) (2.9)

where V is the mutant vector, F and G is the scaling factor with values be-

tween 0 and 1, Xi are randomly selected individuals from the population. The

crossover or recombination method is as follows, regardless of the mutation

method.

U (j ) =

V (j ), if (rand (0,1) < Cr )∪ (j = jrand )

Xi (j ), otherwise

(2.10)

where U is the population member created after recombination. DE may be

an effective algorithm for control system optimization because control system

weighting matrices are real-valued vectors instead of discrete-valued. For bet-

ter final results and faster convergence, the mutation schemes is varied and

tested to see which mutation scheme provides a better final result or finds an

optimal value faster by running system-specific comparisons. A study conduc-

tion in [48] sheds light on the strength of DE compared to GA and PSO.

2.6 Optimal Pareto-front

For multi-objective optimization, with conflicting objectives, there is no single

solution rather a set of solutions. Each solution consists of costs equal to the

number of objectives. One solution is said to strongly dominate another solu-

tion if and only if all the costs of one solution are better than the other or if

and only if all costs are no worse, and at least one is better. If a solution is not

strongly dominated by any other solution then it is a Pareto solution [49]. An

Chapter 2 21

optimal Pareto-front is the set of all such solutions, which are not dominated

by other solutions [50].

2.7 Knee-point

Knee-point is an optimal trade-off point in a set of Pareto-optimal solution.

There are many different methods of calculating the knee-point, and these meth-

ods are studied in detail in [51]. The approach used in this thesis, to determine

knee-point, is the trade-off approach [51]. All the solutions are studied and a

point is chosen, from the knee-region, which has a similar deviation from both

extreme points for each of its costs. The knee-point is discussed and defined in

great detail in [51]. An example of a knee-point is given in Fig. 2.1 where the

point B is the knee-point [51].

2.8 Non-dominated Sorting Genetic Algorithm

NSGA-II (Non-Dominated Sorting Genetic Algorithm) [9] is an evolutionary

algorithm to solve multi-objective optimization problems and get an optimal

Pareto-front solution. Optimal Pareto solutions can be used by the user/re-

searcher to achieve their desired outputs. NSGA-II’s elitist preservation never

discards best solutions [9]. The process of selection mutation and recombi-

nation continues until preset conditions are satisfied and a desired solution

is achieved. The second part of NSGA-II ensures diversity between candidate

solutions. At the end of the entire procedure, an optimal Pareto-front set of

solutions is found.

Chapter 2 22

Figure 2.1: Point B is the knee-point. This image is taken from [51]

Each population member is assigned two values, i.e. ηp , the number of other

solutions which dominate this solution, and Sp , set of solution that the solu-

tion dominates [9]. The process continues until the termination conditions is

reached. Sorting is done based on ηp and Sp , and an optimal Pareto-front is

occupied by the solutions, which has not been dominated by any other solution

[9]. Each solution is visited N −1 times and complexity of the problem is O(N 2)

[9], where N is the population size. The complexity would be O(MN 2), where

M is the number of objectives [9]. Detailed description of the algorithm can be

found in [9].

Chapter 2 23

Crowding Distance, according [9], is defined as a diverse set of solutions pro-

vide a better optimized result. NSGA-II maintains solutions diversity by using

the crowding distance concept. Crowding Distance is calculated by selecting a

solution and then calculating the distance between the point and the two ad-

jacent points in the Pareto-front. Crowding operator sets the desired diversity

among the candidate solutions (crowding distance). Each population member

is assigned a crowding distance and a rank, which corresponds to the rank of

the Pareto-front.

2.9 Generalized Differential Evolution

Non-dominated Sorting Genetic Algorithm (NSGA) is the base of Generalized

Differential Evolution (GDE3). Apart from these two algorithms, there are

other MOEAs, e.g. Pareto Evolution Strategy (PAES) [52] and Strength-Pareto

Evolutionary Algorithm (SPEA) [53]. GDE3 is chosen based on a comprehen-

sive comparison done between these algorithms in [9, 54, 55]. With this in-

formation, GDE3 is compared to NSGA and conventional tuning rather than

re-perform established experiments to gain similar results just to include other

MOEA.

GDE3 removes the GA part of NSGA and replaces it with DE’s method of se-

lection, mutation and recombination. Bit string representation makes GA great

for scheduling and discrete problem, but DE exceeds GA for real-valued prob-

lems [56], e.g. optimization of the LQR controllers. All other aspects of NSGA

are unchanged.

Chapter 2 24

2.9.1 GDE3 Algorithm

In the algorithm 1, D represents number of decision variables, maxit represents

generational iterations, xlb and xub are lower and upper bounds for the search

space respectively, F is the scaling factor, Fmin and Fmax are the lower and

upper bounds of the scaling factor, Np is the population size, V is the mutant

vector. For non dominated sorting, p is the selected individual, Sp are the sets

dominated by p, and np are the number of individuals that dominate p. For

crowding distance, Pfi is the ith pareto front, TPf represents total pareto fronts,

ni represents number of individuals in that pareto front, O1 is value of objective

function 1, and O2 is the value of objective function 2. The GDE3 method is

outlined in algorithm 1.

Chapter 2 25

Algorithm 1 GDE3 Algorithm [57],[9]

1: INPUT : D ,maxit ,Np,xlb ,xub ,Fε(Fmin ,Fmax )2: Initialize population uniform randomly3: Non Dominated Sorting4: for p = 1,p ≤ Np,p++ do5: S (p) = φ6: for q = 1,q ≤ Np,q++ do7: if q == p then8: Skip and check next member9: end if

10: if Np(p)dominatesNp(q) then11: Sp = Sp ∪ q12: elseNp(q)dominatesNp(p)13: np + +14: end if15: end for16: end for17: Pareto-optimal members are such that np = 018: Crowding distance19: Initialize distance between all members of front to be 020: for i = 1, i < TPf , i + + do21: Sort population based on each objectives’ cost22: Assign inf distance to boundary values23: Assign crowding distance based on24: for k = 1,k ≤No. of objectives,k + + do25: for j = 2,k ≤ (ni − 1),k + + do CD(k , j ) =

Oj (k+1)−Oj (k−1)Oj (ni )−Oj (1)

26: end for27: end for28: end for29: Main loop30: while it <maxit do31: Selection32: X1,X2,X3ε{1,2, ...Np}33: jrandε1,2, ...,D34: V = X1 +F ∗ (X2 −X3), Creating mutant vector V35: xlb ≤ V (i ) ≤ xub ,ensure V is within bounds36: Mutation and Crossover:

37: U (j ) =

V (j ), if (rand (0,1) < Cr )OR(j = jrand )Xi (j ), otherwise

38: Selection Xi =

U , , if U dominates Xi

Xi , , otherwise39: it = it + 140: Non-Dominated Sorting41: Calculate Crowding Distance42: it = it + 143: end while

Chapter 3

Testing the Base Algorithm:

Differential Evolution

DE has many variations, mainly due to the difference in mutation schemes. The

effect of changing mutation schemes on control system tuning is the topic of

focus in this chapter. The best mutation scheme variant is found to be DE5,

which makes use of the mutation scheme described in Eq.(2.7).

This chapter also clarifies the effect of a distributed framework on control sys-

tem tuning. It will be examined whether splitting the population into multiple

small islands allowing for information exchange and individual island evolu-

tion has any effect on control system tuning or not. The achieved results show

that distributed framework improves the final results.

It is very challenging and time consuming to test 12 variations of DE on ATS

tuning, for car-trailer combination. This chapter makes use of a PID controller

example to give an overview of the performance difference of DE variants in

control system tuning. There is no guarantee that the best variant in PID con-

troller tuning is the best variant for ATS tuning. The results in further chapters

26

Chapter 3 27

show that the variant selected can tune, within design guidelines.

3.1 PID Controller

We have come a long way in increasing the computational efficiency for com-

puters to run optimization algorithms, which once required highly specialized

equipment. Applications of optimization are in every field of technology, and

every system can be optimized to be better. This section concentrates on tuning

of PID controllers using DE algorithms to obtain better control gain Kp ,Ki ,Kd

values. DE is at the core of GDE3, which will be utilized as a MOEA to tune the

ATS controllers for car-trailer combinations in later chapters of this thesis. To

benchmark the algorithms, a general second order system is considered.

Various methods are proposed for tuning PID controllers. In the view of op-

timization, the goal is to find an optimum solution. Initially, classical opti-

mization techniques were developed to find optimum solutions. Fermat and

Lagrange found calculus-based formulae for identifying optima, while Newton

and Gauss proposed iterative methods for moving towards an optimum. The

term linear programming for certain optimization cases was due to Dantzig,

although much of the theory had been introduced by Kantorovich in 1939 [58].

The classical optimization techniques are beneficial in finding the solution or

unconstrained maxima or minima of continuous and differentiable functions.

Differential calculus is used to reach up to the solution in these types of prob-

lems. As many practical problems are not continuous or differentiable, so clas-

sical methods could not be used for the solution to these types of problems.

Chapter 3 28

However, classical methods are the base for the development of linear program-

ming techniques and many modern optimization methods. Numerical meth-

ods, e.g. Linear Programming, Integer Programming, Quadratic Programming

and Non- linear programming, are used for the solution to many problems.

In this section, the convergence rate and performance of 12 DE variants over 5

different fitness functions (varied by Steady State error calculations) are com-

pared. The objectives are to minimize the following performance measures

of PID controllers: Settling Time, Rise Time, Overshoot and Steady State Er-

ror. The four performance measures make the PID controller turning a multi-

objective optimization problem. In this section, a single objective algorithm is

used with an aggregated fitness function.

3.2 Tuning methods for PID controllers

Conventional PID controllers are categorically the most generally used control

algorithms in various applications due to their practicality. Their compara-

tively simple structures, which can be simply executed, and the accessibility of

well-established rules for tuning the parameters of the controllers are the main

reasons for various real-time applications.

A PID controller is a feedback controller, which continuously calculates the ap-

propriate adjustment whenever the actual condition of a plant differs from the

desired value or set point. It consists of a proportional control term, an inte-

gral control term, and a derivative control term. To get a satisfactory control

performance, one only needs to adjust three parameters of the controller gain,

the integral or rest time, and the rate or derivative time in a PID controller. The

Chapter 3 29

adjustment of these parameters is called tuning, and many experimental meth-

ods have been developed for this purpose, e.g. Ziegler-Nichols and Astrom-

Hagglund [59]. The term P is proportional to the current value of the SP −PV

error, i.e., e(t). Here SP is the set-point and PV is the proposed value or Pro-

cess variable. Term I accounts for the past values of the SP − PV error and

integrates them over time to produce the I term. Term D is the best estimate

for the future trend of the SP −PV error.

Various tuning methods have been proposed from 1942 up to now for gaining

better and more acceptable control system response based on desirable con-

trol objectives, e.g., percent of overshoot, integral of the absolute value of the

error (IAE), settling time, manipulated variable behavior, etc. Some of these

tuning methods have considered only one of these objectives as a criterion for

their tuning algorithm, and some of them have developed their algorithms by

considering more than one of the mentioned criteria [60].

The goal of tuning a PID controller is to make it stable, responsive and to min-

imize overshoot. These goals - especially the last two - conflict with each other.

You must find a compromise between the goals, which acceptably satisfies them

all. Process requirements and physical limitations will determine the balance

between the amount of acceptable overshoot as well as the demand for respon-

siveness. A PID controller is essentially a generic closed-loop feedback mech-

anism. The auto-tuning of PID control started by Ziegler and Nichols. This

method is a trial and error test, in which one must produce oscillations with

constant amplitude. Ziegler and Nichols is an effective conventional technique

for tuning of PID Controllers. However, sometimes it does not provide a good

solution and tends to produce large overshoots.

Chapter 3 30

3.3 DE Algorithm

DE has been benchmarked to be faster and more accurate than genetic algo-

rithms in various research papers [61, 62]. DE uses continuous vectors instead

of chromosome based on bits [63]. Systems, which require population members

to be continuous, e.g., a second order system, greatly benefit from this [62].

A PID controller designed for a second order system has the control gains, Kp ,

Ki and Kd , whose values belong to real continuous numbers. Each population

member of the DE algorithm has three values corresponding to each Kp , Ki ,

and Kd . GA differs by incorporation of various crossover strategies to achieve

better more optimized results, for DE the reliance is on mutation as can be seen

from Eqs.(2.3)-(2.9).

DE algorithm starts with the initialization of the population within the lower

and upper bound defined. Each population member created is evaluated and a

cost is assigned based on the fitness function. The main loop of DE begins after

population initialization. In each iteration of the main loop, three to five pop-

ulation members are chosen at random, depending on the mutation scheme,

and the best member may also be used. Upon these members, the mutation is

performed, according to the selected mutation scheme. Eq.(2.10) shows that

the crossover between current population member and the mutated population

member is performed. Once crossover is completed and a new vector is cre-

ated, which is evaluated against the population member currently selected, if

it is better, it is kept, otherwise discarded. The global best value is always kept,

ensuring elitist preservation. A detailed algorithm is outlined in [46].

Chapter 3 31

As any evolutionary algorithms, DE has a set of parameters. These parame-

ters include Population Size (Np), Crossover probability (Cr ), Scaling Factors

(K&F ), definition of a Search Space (upper and lower bounds), mutation and

crossover schemes. Apart from that, it can have one or more fitness functions

to evaluate how good a population member is and termination conditions, in

case termination condition is based on the number of generations (NFC).

3.3.1 Island distribution of DE

Genetic Algorithm is inspired from real life. Real life is full of information ex-

change and individual or collective evolution. Families evolve, which evolve

cities, which evolves countries, and then there is global evolution. Resources,

information, and technology are exchanged as well. This similar concept can

be seen in the variation of DE, also discussed in [47]. It is called Distributed

Differential Evolution. In this case, the total population is split into islands,

each island has its own individual evolution, and it also sends migrants to ad-

jacent islands as information exchange. This is called Island Based Distributed

Framework with Information Exchange, which is part of a Low- Level Relay

Based Method as well. The same distribution is followed. The Island-based

DE has the following parameters: Number of Islands (ni ), Number of Migrants

(nm), Migration Frequency (mf ), The Migration Topology (mt), the Selection

Policy (sp) and lastly the replacement policy (rp).

In this experiment, Number of Islands (ni ) is 4, Number of Migrants (nm) is

1, and Migration Frequency (mf ) is every 45th generation. Migration topology

(mt) is migration between adjacent islands with the first island not accepting

any migrants. The Selection Policy (sp) for the migrant is selecting the best

Chapter 3 32

member of the island to send to the adjacent island. Replacement policy (rp)

is defined that the migrant will replace any random member, which is not the

best member of the island. Each island can have different mutation schemes,

which allows for high variation. In this thesis, five such variations are consid-

ered: 1) each island uses mutation scheme described in Eq.(2.3); 2) each island

uses mutation scheme defined in Eq.(2.4); 3) each island uses mutation scheme

specified in Eq(2.5); 4) each island uses mutation scheme expressed in Eq.(2.6);

and 5) island 1 uses Eq.(2.3), island 2 uses Eq.(2.4), island 3 uses Eq.(2.5), and

island 4 uses Eq.(2.6). Table 3.1 lists all DE labels and their definitions.

Table 3.1: The table lists the assigned labels of the variants of DEbased on mutation schemes.†

Given Label Mutation Strategy Distributed (Yes/no)DE1 Eq.(2.3) NoDE2 Eq.(2.4) NoDE3 Eq.(2.5) NoDE4 Eq.(2.6) NoDE5 Eq.(2.7) NoDE6 Eq.(2.8) NoDE7 Eq.(2.9) NoDE8 Eq.(2.3), on all Islands YesDE9 Eq.(2.4), on all Islands Yes

DE10 Eq.(2.5), on all Islands YesDE11 Eq.(2.6), on all Islands Yes

DE12

Eq. 2.3, on Island 1 YesEq.(2.4), on Island 2Eq.(2.5), on Island 3Eq.(2.6), on Island 4

† For example DE4 is the label assigned to a variant of DE which usesEq.(2.6) as the mutation scheme. These labels are used in the analysissection.

Chapter 3 33

3.4 Objectives

When tuning a system, many things must be considered. Settling Time, Rise

Time, Overshoot are among some of them, as well as Steady State Error. If all

four of these are considered as fitness criteria, then it is a multi-objective op-

timization problem. A fitness function, which is an aggregate of all four, can

be a good approach in finding an optimal solution. The final solution needs to

have acceptable values for Kp , Ki , and Kd , which will result in the minimiza-

tion of all four objectives. In this section, the four objectives are dealt with by

aggregating them using Eq.(3.1) [64]. The aggregation approach outlined in

[64] is used. This approach utilizes a varying weighted fitness function. The

simple aggregation method is adding all the objectives together and convert-

ing a multi-objective into a mono-objective or single objective problem. This

did not result in a great variation when the system is tuned, nor a good set of

values. The approach used is followed from [64], where Eq(3.1) is used to get

better results.

F =1

1 + e−α∗ (RT + ST ) +

e−α

a + e−α∗ (OS + SS ) (3.1)

Where RT is the Rise Time, ST is the Settling Time, OS is the Over Shoot, SS

is the Steady State Error and F is the fitness/objective function. Here α is the

weighted variable. The value of α will vary in the range [-5, 5] [64].

Chapter 3 34

3.5 Steady State Error

RT , ST , and OS are generated through simulation graphs directly, but steady

state error must be calculated. To calculate SS , there are various techniques,

which are described in Eq.(3.2)-(3.6). The methods to find SS are taken from[65]

for comparison. Adding a variety of SS methods in simulation ensures a reli-

able comparison and a more concrete conclusion.

MSE =1t∗ˆ

e(t)2dt (3.2)

ITAE =ˆ

t ∗∣∣∣e(t)

∣∣∣dt (3.3)

IAE =ˆ ∣∣∣e(t)

∣∣∣dt (3.4)

ISE =ˆ

e(t)2dt (3.5)

ITSE =ˆ

t ∗ e(t)2dt (3.6)

where MSE (Mean Squared Error), ITAE (Integral of Time Multiplied by Ab-

solute Error), IAE (Integral of Absolute Magnitude of the Error), ISE (Integral

of the Squared Error) and ITSE (Integral of the time multiplied by Squared Er-

ror) are the five methods to calculate SS listed in [65]. Eq.(3.1) is the fitness

function with a steady-state error component. Five different ways to calculate

steady state error results in five different variants of the fitness functions. This

allows for the comparison of the performance of not only DE for tuning, but

how variation in Steady State Error calculation affects DE performance as well

as the final response of the second order system.

Chapter 3 35

3.6 Second Order System

A system whose input-output equation is a second order differential equation

is called Second Order System. There are many factors that make second-order

systems important. They are simple and exhibit oscillations and overshoot.

Higher order systems are based on second order systems. The second order

systems being used as a test bench is defined in Eq.(3.7) and the controller in

Eq.(3.8).

1s2 + 10s + 20

(3.7)

kp +kis

+ kd ∗ s (3.8)

3.7 Performance Comparison

The first experiment deals with the performance of DE. Variants of DE are com-

pared, for the given parameters. The results of the experiment show the advan-

tages of islands and hybridization along with which mutation scheme outper-

forms the others. Steady State Error is omitted from Table 3.4, as it is spread

over response time. Rise-Time, Settling-Time and Peak-Overshoot, listed in Ta-

ble 3.4, are used to ably judge the system performance.

The DE parameters are: Np = 48, Cr = 0.95, Generations = 200, and the number

of runs to average values = 5. Moreover, the search space is [1, 250]. We also

tuned the same system using MATLAB built-in PID Tuner and gained values

of Settling Time, Response Time and Overshoot, as well as a random value to

compare DE. A random value is used to ensure that the algorithm is working

using evolution and not due to randomness. These two are anchors around

Chapter 3 36

Table 3.2: The experimental results for DE1 to DE6 and the average RT, STand OS.

SS PM DE1 DE2 DE3 DE4 DE5 DE6MSE RT(s) 0.1589 0.0854 0.1695 0.1490 0.1441 0.1159MSE ST(s) 0.2620 0.9777 0.3896 0.4027 0.2294 0.7359MSE OS 0 0 0 0 0 0

ITAE RT(s) 0.1366 0.1239 0.1510 0.1482 0.1672 0.1596ITAE ST(s) 0.6261 0.6795 0.2346 0.2817 0.3183 0.2525ITAE OS 0 0 0 0 0 0

IAE RT(s) 0.1322 0.1530 0.1407 0.1375 0.1359 0.1482IAE ST(s) 0.1844 0.2444 0.2140 0.1981 0.2015 0.2095IAE OS 1.3798 0 0 1.1198 0.4546 1.8158

ISE RT(s) 0.1567 0.1468 0.1724 0.1361 0.1358 0.1566ISE ST(s) 0.2438 0.3449 0.2851 0.2021 0.2055 0.3030ISE OS 0.0494 0 0 0.4220 0.0322 0

ITSE RT(s) 0.0920 0.0912 0.0895 0.0900 0.0891 0.0916ITSE ST(s) 0.9034 0.9080 0.9195 0.9165 0.9222 0.9055ITSE OS 0 0 0 0 0 0

AVG RT(s) 0.1353 0.1201 0.1446 0.1322 0.1344 0.1344AVG ST(s) 0.4447 0.6309 0.4086 0.4002 0.3754 0.4813AVG OS 0.2858 0 0 0.3083 0.0974 0.3631

which the algorithm performs. The experiment is repeated 51 times and the

values averaged to eliminate stochastic bias in evolutionary algorithms.

3.7.1 Results and Analysis

Tables 3.2 and 3.3 offer the results of the experiments. Table 3.4 displays the

results from MATLAB PID Tuner and a random value. The purpose of the ran-

dom value is to emphasize the convergence of the evolutionary algorithm. A

comparison of the results offered in Tables 3.2 and 3.3 with those provided in

Chapter 3 37

Table 3.3: The experimental results for DE7 to DE12 and the average RT, STand OS.

SS PM DE7 DE8 DE9 DE10 DE11 DE12MSE RT(s) 0.1489 0.1258 0.0975 0.0992 0.1343 0.1458MSE ST(s) 0.5434 0.6719 0.8900 0.8608 0.2463 0.2510MSE OS 0 0 0 0 0 0

ITAE RT(s) 0.1652 0.1480 0.1415 0.1333 0.1310 0.1571ITAE ST(s) 0.3535 0.2513 0.2194 0.6408 0.5903 0.2732ITAE OS 0 0 0 0 0 0

IAE RT(s) 0.334 0.1305 0.1355 0.1332 0.1560 0.1679IAE ST(s) 0.1957 0.2050 0.2052 0.2158 0.3084 0.2890IAE OS 0.6483 0 0.0703 0 0 0

ISE RT(s) 0.1342 0.1714 0.1438 0.1398 0.1466 0.1362ISE ST(s) 0.1967 0.2682 0.3963 0.2101 0.2066 0.2576ISE OS 0.6645 0.4475 4.9219 0.2383 1.7173 0

ITSE RT(s) 0.0893 0.0900 0.0879 0.0912 0.0895 0.0902ITSE ST(s) 0.9210 0.9160 0.9309 0.9083 0.9198 0.9150ITSE OS 0 0 0 0 0 0

AVG RT(s) 0.1342 0.1322 0.1213 0.1193 0.1315 0.1394AVG ST(s) 0.4421 0.4625 0.5283 0.5672 0.4543 0.3972AVG OS 0.2626 0.0895 0.9984 0.0477 0.3435 0

Table 3.4: MATLAB PID tuner and random value results

Method RT(s) ST(s) OvershootPID Tuner 0.3523 1.2112 6.3913

Random Value .9355 2.1074 0.6770

Table 3.4 reveals that the all DE variants show better performance than the Mat-

lab PID Tuner and Random Value for the gains. The difference is not minute.

This shows that the DE variants can, regardless of the mutation scheme, per-

form better than conventional methods for PID tuning. In terms of the rising

time, DE10 performs the best, DE5 achieves the shortest settling time, and DE2,

DE3, and DE12 all can achieve 0 overshoot. From this experiment, it can be

Chapter 3 38

Figure 3.1: Responses for DE1 and MATLAB Tuner and Random Value

concluded that DE may greatly enhance the performance of the Second Order

System when compared to other conventional methods. Table 3.3 displays that

Island-based DE and Hybrid DE perform very well. DE10 and DE12 are Island-

based and hybrid DE, respectively, and they can optimize 2/3 of the parameters

better than their counter parts. With further optimization and tweaking of the

Islands and Hybrid Algorithm or increasing the number of runs, better results

can be achieved. Figs. 3.1-3.2 show the responses for DE1 and DE2. The re-

sponses of other DE variants are similar.

Chapter 3 39

Figure 3.2: Responses for DE2 and MATLAB Tuner and Random Value

3.8 Convergence Speed Comparison

Convergence speed is an integral factor when tuning control systems. Conver-

gence speed is defined by how fast does an evolutionary algorithm reach the

optimal point. The best fitness value is noted every NFC/Np generations and a

logarithmic plot is created. Each variant of DE is plotted and ranked based on

how quickly the best fitness value is achieved. The ranks go from 1 through 12,

1 being the best and 12 being the worst, and each variant is ranked for all five

SS error calculation methods.

Best rank meant that DE can find the best solution faster than any other in the

competition. In case of a tie, both variants are given the same rank, and the

next rank is skipped. The experiment is run 51 times, and the final averages

are used to determine rank to eliminate stochastic bias.

Chapter 3 40

Table 3.5: Rank of DE Variants in Convergence Comparison

Variant IAE ISE ITAE ITSE MSE AVGDE1 10 10 10 9 11 10DE2 12 9 3 7 8 7.8DE3 11 7 5 5 5 6.6DE4 7 5 1 6 9 7.6DE5 1 2 2 1 1 1.4DE6 9 8 9 12 10 9.6DE7 2 3 1 8 2 3.2DE8 6 6 8 4 12 7.2DE9 8 12 12 10 4 9.2

DE10 5 11 7 3 7 6.6DE11 3 1 4 2 3 2.6DE12 4 4 6 11 6 6.2

3.8.1 Results and Analysis

Table 3.5 shows the average rank of each DE variant over the 5 Steady State

Error methods. From Table 3.5, it is observed that DE5 is the best, followed

by DE11. This comparison focuses on the convergence rate of the fitness func-

tion in Eq.(3.1) rather than individual responses (i.e., ST, RT, OS). This graph

sheds great light onto the performance of various DE mutation schemes and

which achieve the best results. On Average Non-distributed, DE methods reach

a rank of 6.6, as compared to 6.36 of Distributed DE. One aspect which skewed

result is DE5, which performs the best, is not part of the distributed mutation

schemes.

3.9 Summary

It is apparent, from the results, that mutation scheme and distribution have an

impact on control system tuning using DE. The overall best mutation scheme

Chapter 3 41

is DE5, which is described in Eq.(2.7). Though distribution brings an improve-

ment in the final results, the improvement is not considerable enough to add a

distributed framework to GDE3. The major purpose of this chapter is achieved,

and the DE5 variant is henceforth used in GDE3 for control system tuning in

further chapters.

Chapter 4

GDE3 and NSGA-II Comparison on

LQR Based Aircraft Pitch Control

Generalized Differential Evolution (GDE) has not been used, to the best of the

authors knowledge, to tune the controllers designed using the LQR technique.

It is important to test a new algorithm on a well-defined benchmark case. The

case study on aircraft pitch control is taken from [66] for this purpose. It has

a clear definition, design goals, and implementation. GDE3 will be compared

to conventional tuning methods as well as its genetic algorithm counterpart,

NSGA-II. The contents of this chapter have been published in IEEE Congress

on Evolutionary Computation 2019. The case study shows that GDE3 outper-

forms NSGA-II and other conventional tuning methods, and GDE3 outputs an

optimal Pareto-front closer to the origin for this minimization problem

42

Chapter 4 43

4.1 Introduction

A combination of two existing techniques are used to prove that the resulting

algorithm performs better than the parent techniques or conventional meth-

ods. The parent optimization algorithm used in this paper is the DE [46]. For

the current study, two conflicting objectives and multiple trade-off solutions

are required to be found. Based on the situation, the designer, as a decision

maker (DM), can choose the most appropriate solution. So enhancing LQR con-

troller can be defined as a multi-objective optimization problem. Evolutionary

computation (EC), being a powerful method, has been used to solve many real-

world multi-objective optimization problems [67, 68]. The contribution of this

chapter is providing a system designer with a set of optimal solutions for var-

ious conditions, which satisfy design requirements and conditions. In order

to achieve an optimal Pareto-front, a variation of Non- dominated Sorting Ge-

netic Algorithm (NSGA-II) is employed [9], which is Generalized Differential

Evolution (GDE3). GDE3 uses differential evolution (DE) instead of genetic

algorithm (GA).

Aircraft pitch control is used as the case study to compare the performance of

GDE3 and NSGA-II, and to show the advantages of multi-objective optimiza-

tion. The aircraft model and corresponding results are from a collaborative

controls tutorial set-up by University of Michigan, Carnegie Mellon University,

and University of Detroit Mercy [66]. GDE3 results are compared to the results

obtained from [66] and NSGA-II, while keeping the same model to ensure fair

comparison and obtain a concrete conclusion. The existing model from [66] is

Chapter 4 44

run through the GDE3 optimizer to generate a set of optimal Pareto-front so-

lutions. The optimizer ensures that the entire solution set is stable by ensuring

that every value results in a system, which settles to 0. The dynamic responses,

i.e., settling-time, rise-time and overshoot, are measured, for some values in the

solution set, and compared to the response from [66] and NSGA-II.

The novelty of the proposed method is that it provides the system designer with

an option when configuring an LQR controller. The method goes beyond the

conventional optimization of using a weighted aggregation of multiple objec-

tives, and results in a solution, which works but can be made much better. This

case study demonstrates this hypothesis.

4.2 System Model

Evolutionary algorithms, by nature, are slow. Real-time systems, on the other

hand, require faster algorithms to generate real-time results, which renders op-

timization unrealistic for online tuning. For change in each situation, the sys-

tem has to be re-optimized after the weights are tuned, to get a new singular

solution. The proposed approach provides a massive advantage against this

traditional strategy. For LQR based systems, the proposed method provides

the system designer an optimal set of solutions resulting from multiple design

objectives. All of these solutions can be stored as a look-up table, with values

based on environmental inputs. The LQR-based control system continues to

function, as it would otherwise, while the decision system reads the environ-

mental inputs. Based on the environmental inputs, the decision system selects

appropriate solution to recalibrate the LQR gain in real-time.

Chapter 4 45

Fig. 4.1 shows how this model works. This process is completely deterministic

as all the solutions and their respective responses are already calculated offline.

Similarly, all the environmental conditions, in which the gain would change, is

predetermined by the system designer when creating the look-up table. The ex-

ecution time analysis can be done on the system beforehand, to ensure it meets

all standards. Active steering system for Articulated Heavy Vehicles (AHV) [3]

is an example of a real-time system using an LQR controller [39, 69].

An active steering system is designed to improve the lateral stability at high

speeds to ensure that the vehicle does not rollover [39, 69]. Similarly, it must

enhance the low-speed maneuverability so that the vehicle can safely negoti-

ate 90-degree turns at intersections. Both of these are conflicting objectives

[3].Once optimal Pareto solutions are obtained, they can be loaded onto the

decision system. Based on environmental requirement, the LQR gain can be

changed using pre-optimized stable solutions. From Fig. 4.1, it can be ascer-

tained that this method can be applied to any system with an LQR controller.

4.3 Multiple Objectives

Real-time systems must adhere to many restrictions, and achieve a variety of

performance measures or objectives. Minimization of rise-time (RT) and settling-

time (ST) are examples of two common conflicting objectives in control systems,

which are incorporated in this case study. Evolutionary Algorithms, in their

bare form, require a single fitness function based on one or many objectives.

In other works, multiple objectives have been aggregated, using weights, into a

single fitness function, which is then optimized to produce a single optimized

Chapter 4 46

Figure 4.1: Feedback system model with LQR and decision support system.Open-loop plant is the aircraft pitch model. Controller gains are obtained bythe algorithm, which can be dynamically changed by the decision system based

on environmental factors.

solution [64]. A single aggregated weighted fitness function cannot achieve the

convex part of the Pareto-front, otherwise, multiple varying weights might be

required, which add a layer of complexity to it [9]. Secondly, assigning weight

has to be completed before the algorithm begins. Search for a better result is

not conducted after finding a satisfactory result. Posteriori decision-making

can find a better solution, even though an acceptable solution is found. After

completing the optimization process, both GDE3 and NSGA-II provide a set of

optimal Pareto-front solutions [9]. The decision-maker can choose any solution

that best fits the system, without having to rerun the experiment by adjusting

weights assigned to each objective. A common example, the selection of airfare

between point A and point B has two objectives: cost and time. If a ticket costs

more, it has fewer connections, and less travel time will spend, and vice versa

Chapter 4 47

(generally). Cost and time, in the case of airfare, are conflicting objectives.

4.4 Ensuring Stability and Controllability

The Q weighting matrix (see Eq.(2.2)), in a LQR controller design, must be

positive semi-definite, that is, all their eigenvalues must be greater or equal to

0 [70]. This algorithm ensures that for each iteration the generated Q matrix

meets the criteria. In this experiment, the R matrix is set to 1, and the elements

of the Q matrix are varied to tune the LQR controller. The optimizer ignores

any selection, which cannot stabilize the system, and this ensures that all final

solutions are stable.

In order to ensure that a system model is compatible with this method, it must

be stable, observable, and controllable. In this case, the system model refers to

the state space representation of a particular problem, including the selection

of design parameters. The most common check for system stability is to check

the poles; a dynamic system must have a finite number of non-negative poles to

be controllable. One way to do this, in MATLAB, is to use the command isstable,

which returns a value of 1 (true) if the system is stable and 0 (false) if the sys-

tem is unstable. If the real part of the eigenvalues of matrix A is negative values,

then the system can be deemed as asymptotically stable. Negative eigenvalues

show that the system will tend to move back to the steady state. Controlla-

bility and observability can be checked to ensure the states of the system are

controllable and observable. Controllability refers to how much the input can

be controlled by manipulating the state. Observability indicates how much can

be perceived from the output. In order to calculate Controllability, the rank of

Chapter 4 48

Table 4.1: Control parameters of Optimization

Parameter ValuePopulation Size (P ) 60

Crossover Probability (Cr ) 0.95Scaling Factor (F ) 0.3 < F < 0.9

Number of Objectives 2Range of xi 0.01 < x < 10

[A...An−1B ] must be equal to the size of the Matrix A, where the matrix B is the

input-to-state matrix (see Eq.(2.1)). For calculating observability, the rank of

[CT ...CAn−1T ] has to be equal to the size of the matrix A, where the matrix C is

the state-to-output. All the states in the state space models are controllable and

observable, this is ensured by using minimal realization or pole-zero cancella-

tion [66]. Each gain matrix obtained by the multi-objective tuning algorithm

is simulated and tested to ensure that the output settles to origin and is within

the boundary requirements, this ensures output stability.

4.5 Optimization Parameters and Pre-compensation

For the aircraft pitch control using LQR, the same DE parameters are used,

which are listed in Table 4.1. The parameters listed in Table 4.1 are obtained

from [47], in which the authors do rigorous testing to show that these produc

better results. The experiments has been run for 51 times to minimize the im-

pact of stochastic effects on the reported results.

When a full-state feedback system is created, the actual output of the system

and the desired output as seen in the actual model can vary and requires scaling

[66]. The output received by the optimal controller is not compared to the

reference, rather it compares States ∗Kx to the reference [66]. This is done as per

Chapter 4 49

the full-state feedback system instructions in [66], to ensure a fair comparison.

After the results are obtained, the pre-compensation procedure listed in [66] is

followed to scale the output of the control system to match the desired output.

The scaling factor does not affect the objectives, by using pre-compensation,

and fair comparison among competing algorithms is ensured.

4.6 Complexity of Case Study

The Aircraft Pitch Control is a complex case study. The case study is chosen

as it is similar to the ATS controller design problem tackled in the thesis. Two

aspects determine the complexity of the system: size of the system matrix A

and decision variables in gain matrices Q and R. The system matrix in the

case study is 3 ∗ 3 and decision variables in Q and R are nine. An adaptive

cruise control has been modeled by a 3∗3 system matrix [71] with nine decision

variables and a simple cruise control can be modeled as a 1 ∗ 1 system matrix

with 1 decision variable [66]. An active steering system for an articulated heavy

vehicle can be modeled by a 4 ∗ 4 system matrix with 4 decision variables [25,

72]. A two-wheel self-balancing robot is modeled by a 4 ∗ 4 system matrix [73]

with 2 decision variables. A quarter car suspension system can be modelled by

a 4 ∗4 matrix with 4 decision variables [17, 66]. This shows that the complexity

of the selected systems matches or exceeds many other comparable systems.

4.7 LQR Controller for Aircraft Pitch Control

An aircraft in flight is a complex dynamic system and may exhibit three trans-

lation and three rotational motions [74]. The case study focuses on one of these

Chapter 4 50

rotational motions of the aircraft, i.e., pitch around the lateral axis [74]. To have

an effective comparison, the aircraft model given in [66] is used to optimize and

investigate the method’s performance. The state space model for the aircraft is

defined by Eqs.(4.1) and (4.2). The model used in [66] assumes that the aircraft

is in steady-cruise at constant altitude and velocity, the thrust, drag, weight and

lift forces balance each other in x and y directions, pitch angle does not affect

the speed of the aircraft. Implementing an LQR as a feedback controller does

not skew comparison results as it is same for the both cases. The input is a step

reference of 0.2 rad, where the following criteria must be met: overshoot less

than 10%, rise-time less than 2 seconds, settling time less than 10 seconds [66].

The minimization objectives are rise time and settling time. Fig. 4.2 shows

the full states-space feedback control for aircraft pitch control used in the case

study [66] with the decision system, where EI corresponds to environmental

inputs that affect the decision-making process, for example considering the air-

craft speed. Where x = [α,q ,θ]′ is the state vector, θdes is the reference (r ),

δ = (θdes −Kx ) is the control input (u), θ is the output (y) and K is the control

gain matrix. K is obtained by the LQR method in the case study.

f ′(x ) =

−0.313 56.7 0

−0.0139 −0.426 0

0 56.7 0

α

q

θ

+

0.232

0.0203

0

[δ

](4.1)

y =[0 0 1

]α

q

θ

+[0

][δ

](4.2)

Chapter 4 51

Q =

x1 x2 x3

x4 x5 x6

x7 x8 x9

(4.3)

4.7.1 Experimental Results

Fig. 4.3 shows the optimal Pareto-front obtained by GDE3 and NSGA-II. Each

point is a candidate solution with two objective values, ST and RT. Each solution

corresponds to a control gain matrix of the LQR controller. The decision maker

can pick any solution from the optimal Pareto set, each solution is stable and

optimized and gained in a single optimization run. Figs. 4.4, 4.5, 4.6 the show

responses with least settling time, least rise time and knee-point, respectively.

All solutions are well fitted within the design requirements. Fig. 4.7 is the

response, after conducting optimization process from [66].

Figure 4.2: Feedback system model for aircraft pitch control with GDE3

Chapter 4 52

Figure 4.3: Pareto-front obtained by GDE3 and NSGA-II on aircraft pitch con-trol model. (Note: The designer can choose any LQR gain matrix value from

the Pareto-front, based on ST and RT requirements.)

Figure 4.4: Response of the air-craft pitch control system with ST

prioritized.

Figure 4.5: Response of the air-craft pitch control system with RT

prioritized.

Chapter 4 53

Figure 4.6: Response of the air-craft pitch control system with

knee-point.

Figure 4.7: Response of the air-craft pitch control system ob-

tained from [66].

4.7.2 Analysis

Table 4.2 proves the initial hypothesis. NSGA-II outperforms conventional tun-

ing methods but GDE3 outperforms both NSGA-II and conventional tuning in

RT, ST, and Overshoot at both extremes and knee points. Table 4.2 also shows

that none of the points violated the design constraints. GDE3 is 75.9% better

than conventional and 30.9% better than NSGA-II for RT at knee-point. GDE3

is 81.6% better than conventional and 33.7% better than NSGA-II for ST at

knee-point. The same pattern continues for the extreme points as well. The

best RT achieved by GDE3 is 76.6% better than conventional and 25.7% better

than NSGA-II. The best ST achieved by GDE3 is 87% better than conventional

and 52% better than NSGA-II. The best overshoot achieved by GDE3 is 59.3%

better than conventional and 7.3% better than NSGA-II. The knee-point is cho-

sen in the comparison, as it reflects an aggregated fitness function with equal

weights assigned to both objectives, which is a common scenario. As observed,

both evolutionary algorithms perform better than conventional method. From

Fig. 4.3, it is apparent that all non-dominated solutions of GDE3 dominate the

non-dominated solutions of NSGA-II.

Chapter 4 54

Table 4.2: Comparison Aircraft Pitch Control

Method RT(s) ST(s) Overshoot (%)Conventional Tuning [66] 0.7283 2.0181 4.9100

NSGA-II Knee-point 0.2537 0.5605 2.3425NSGA-II ST 0.2570 0.5467 2.1578NSGA-II RT 0.2295 0.8621 6.4460

GDE3 Knee-point 0.1753 0.3718 2.1369GDE3 ST 0.1754 0.2625 2.0000GDE3 RT 0.1706 0.4595 3.8928

4.8 Summary

This is an important case study before using the algorithm in expensive opti-

mization of the ATS controller for car-trailer combination as this proves the via-

bility of GDE3. In Chapter 5, the same algorithm is utilized to optimize the ATS

controller for a car-trailer combination, but there is no reference design avail-

able for comparison to gauge the improvement that is brought by the GDE3

algorithm. Using an existing well-established case study, with complete exper-

imental detail available in [66], this section proves the practicality of MOEA

algorithms, e.g. GDE3 and NSGA-II.

The GDE3 provides both an optimized result and a solution set to allow for

online decision-making in real-time systems. The experimental results for air-

craft pitch control prove that multi-objective optimization of an LQR controller

is possible, providing both choice and better results than conventional meth-

ods and NSGA-II. This method can be applied to any physical model, which

can be mathematically represented and can be controlled by LQR integration.

As long as the designer can provide an accurate state space representation and

conflicting objectives, this scheme can be used to tune the LQR controller by

manipulating Q and R matrices. These matrices will change with the physical

Chapter 4 55

model as would the complexity of the problem.

Stability and controllability of the physical model are ensured. The algorithm

is a not model oriented scheme and hence, by inference, should provide similar

results for well defined, real-time systems. The aircraft pitch control provids

a real-time example. This chapter of the thesis concentrates on proving GDE3

as a viable method for offline optimization for LQR controller in safety-critical

systems, which not only improves system performance, but also allows for on-

line selection of optimized solution based on environmental factors. . In the

next chapters, these techniques are applied to the articulated vehicles for ATS

controller design.

Chapter 5

Design Optimization of LQR-based

ATS Controller Using GDE3

Chapter 3 and 4 serve as the foundation for the design optimization of the LQR-

based ATS controller for car-trailer combination using GDE3. This chapter uti-

lizes GDE3 to tune the LQR-based ATS controller a car-trailer combination rep-

resented by a 3-DOF yaw-plane model. The tuning is carried out at varying

speeds, and for each speed, an optimal Pareto-front is generated. Each point in

the Pareto-front cluster corresponds a control gain matrix K . This point cluster

may serve as a look-up table for the gain scheduling controllers. The contents

of this chapter have been published in SAE2019 World Congress [72].

5.1 Introduction

This chapter presents a robust ATS controller for car-trailer combinations. ATS

systems have been proposed and explored for improving the lateral stabil-

ity and enhancing the path-following performance of car-trailer combinations.

56

Chapter 5 57

Most of the ATS controllers were designed using the LQR technique. In the

design of the LQR-based ATS controllers, it was assumed that all vehicle and

operating parameters were constant. In reality, vehicle and operating parame-

ters may vary, which may have an impact on the stability of the combination.

For example, varied vehicle forward speed and trailer payload may impose neg-

ative impacts on the directional performance of the car-trailer combination.

Thus, the robustness of the conventional LQR-based ATS controllers is ques-

tionable. To address this problem, a LQR-based ATS controller with a look-up

table gain-scheduling is proposed. In the design of the proposed ATS controller,

at each operating point, the ATS controller is designed using the LQR control

technique. At an operating point between two established adjacent operating

points, the control gain matrix of the controller is determined using a linear

interpolation method. To further improve the directional performance of the

car-trailer combination, the weighting matrices of the LQR controller are de-

termined using an optimization algorithm, namely, GDE3. The effectiveness of

the proposed ATS controller is demonstrated using numerical simulation based

on a car-trailer model.

5.2 Rearward Amplification (RWA)

Each AV may experience different operating conditions, e.g., traveling on a

highway at a high speed or negotiating a curved path at a low speed. From

the view of vehicle design, the criteria are different under the aforementioned

operating conditions. When the AV is travelling at high speeds, it requires to be

laterally stable. In the case of losing lateral stability, an AV may experience one

Chapter 5 58

of the following unstable motion modes: trailer sway, jackknifing, and roll-over.

To evaluate the lateral stability of an AV at high speeds, rearward amplification

(RWA) is a well-accepted performance measure [75].

The RWA measure may be determined under an evasive maneuver at a high

speed. During an evasive maneuver, e.g., a single lane-change maneuver, the

RWA may be defined as the ratio of the peak lateral acceleration observed at the

center of gravity of the trailer and the leading vehicle unit [43]. An ideal RWA

measure is 1.0, which is extremely difficult to achieve in real-world operating

conditions. Physically, when the RWA takes the value of 1.0, the trailing vehicle

unit will have similar dynamic behaviors as the leading vehicle unit. In other

words, with the RWA measure of 1.0, the AV will have better path-following

capability, and the driver may well control the vehicle to achieve high lateral

stability [43]. Controlling lateral acceleration amplification and achieving the

RWA measure of 1.0 is the primary objective. In addition, controlling the RWA

ensures a higher lateral acceleration considering the rollover threshold value.

Articulated vehicles tend to have a low rollover threshold, as low as 0.6g [37].

Since the RWA is evaluated in the linear dynamics of the vehicle, under testing

evasive maneuvers, the peak lateral acceleration of the leading vehicle unit is

set around 0.30g . Equation (5.1) is used to calculate the RWA ratio:

RWA =| Peak lateral acceleration of the trailer at CG || Peak lateral acceleration of the car at CG |

(5.1)

Chapter 5 59

5.3 Path-Following Off-Tracking

At low speeds, the trailer must follow the same trajectory as the leading vehi-

cle unit. If the trailer is unable to follow the same path as the car, there is a

great chance for the trailer to violate the boundary of the road. The measure

to check the trailer ability to track the path of the car is called Path-Following

Off-Tracking (PFOT) [76]. PFOT is defined as the maximum radial offset be-

tween the path of the center of the car front axle and that of the center of the

trailer rear axle over an evasive maneuver. Accurate measurement of PFOT can

be done by comparing the difference during a circular motion of an AV. If a

controller is optimized for high-speed RWA, PFOT performance at low speeds

may be degraded and vice-versa [3]. This makes this problem an optimization

problem, where the parameters of the ATS controller are optimized.

5.4 Design ATS Controller

The design of the ATS controller involves modelling of the car-trailer combina-

tion and the implementation of the control system. To evaluate the resulting

ATS system, a single lane-change maneuver is simulated. The proposed ATS

system is designed based on a few performance measures, e.g., the lateral ac-

celeration of the leading unit and the RWA of the combination.

A multibody dynamic software package, known as Equation of Motion (EOM)

[77], is used to derive the linear 3-DOF yaw-plane car-trailer model. Once the

derived model is verified, EOM is used to develop and simulate the ATS perfor-

mance. The main purpose of using the ATS system is to control and reduce the

Chapter 5 60

lateral acceleration of the combination to ensure that the trailers lateral accel-

eration is not amplified compared to that of the car. In other words, a reduction

in the RWA value will lead to a better dynamic performance of the combination

[78].

The control method used in this study is known as LQR. The LQR-based ATS

controller is optimized using GDE3, which is a MOEA. The LQR-based ATS

controller and GDE3 will be discussed in the following sections.

5.4.1 3-DOF Yaw-plane Car-trailer Model

In this research, the mathematical model is derived and validated with the pub-

lished models [5, 37]. In this model, each axle is represented by a single wheel,

assuming that both tires on each axle have the same dynamic characteristic,

i.e., the relationship between the tire slip angle and the cornering force. Fig.

5.1 shows the schematic representation of the car-trailer combination using the

3 DOF yaw-plane model.

For the yaw-plane model, the lateral and yaw motions of the car, as well as, the

yaw motion of the trailer are considered. The governing equations of motion of

the car are expressed as:

m1(U −Vr ) = −X1cosδ −X2 +X (5.2)

m1(V +Ur ) = f1(α1) + f2(α2) +X1sinδ −Y (5.3)

I1r = af1(α1)− bf2(α2) +αX1sinδ+ dY (5.4)

Eq.(5.2) to (5.4) govern the longitudinal, lateral, and yaw motions of the car,

Chapter 5 61

Figure 5.1: Schematic representation of the 3-DOF yaw-plane car-trailermodel

respectively. Where m1 is the mass of car, U is forward speed of car, V lateral

speed of car and r yaw rate of car, δ is the front wheel steering angle, Xi are the

longitudinal tire forces, αi are the slip angles of the tires, fi are the cornering

stiffness of the tires, X is the longitudinal hitch force, Y is the lateral hitch

force, I1 is the moment of inertia of car and a, b, c and d are described in Table

5.1.

Similarly, the governing equations of motion for the trailer are shown as fol-

lows:

m2(U′−V

′r′) = −X3 −Ysinψ −Xcosψ (5.5)

m2(V′+U

′r′) = f3(α3) +Ycosψ −Xsinψ (5.6)

I2r′= −hf3(α3)− e(−Ycosψ +Xsinψ) (5.7)

Eqs.(5.5) to (5.7) govern the longitudinal, lateral, and yaw motions of the trailer,

Chapter 5 62

respectively. Where m2 is the mass of trailer, U′

forward speed of trailer, V′

lateral speed of trailer, r′

is the yaw angle of trailer, I2 is the moment of inertia

of trailer and h and e are described in Table 5.1.

Once a mechanical hitch is introduced, the kinematic constraint is active. To

simplify the model, the forward speed of the car is assumed constant. In ad-

dition, the articulation angle ψ assumed to be small, which leads to Eq.(5.8) to

(5.10). Detail is available in [72].

cos(ψ) ≈ 1 (5.8)

sin(ψ) ≈ ψ (5.9)

Also, the following equation is determined at zero initial conditions.

ψ = r − r′

(5.10)

Where ψ is the articulation angle, r is the yaw rate of car and r′

is the yaw

rate of the trailer. The notation is shown in Figure 5.1, and the values of the

parameters used throughout this study are listed in Table 5.1.

Once the model is derived, EOM software package is used to validate the model

by comparing the responses of both models under the same single lane-change

maneuver. Both models generate the identical dynamic response, which proves

that the EOM model matches the derived equations. Thus, the EOM software

package is selected to design the ATS controller. However, in order to further

enhance the model, more essential components are added to the system, e.g.,

an actuator is installed on the trailers axle, and an accelerometer is installed at

Chapter 5 63

Table 5.1: Car-trailer combination parameters for ATS controller design

Name Symbol ValueCar Mass m1 1700kg

Car yaw inertia I1 2000kgm2

Trailer mass m2 2000kgTrailer yaw inertia I2 3000kgm2

Distance between CG and front axle a 1.5mDistance between CG and rear axle b 1.7m

Distance car CG to contact with trailer d 2.9mDistance trailer CG to contact with car e 3.8m

Distance between the CG and the rear axle h 0.2mHeight of vehicle CG H1CG 0.325mHeight of trailer CG H2CG 0.676m

Combined front tires cornering force c1 80000Nmrad

Combined rear tires cornering force c2 80000Nmrad

Combined trailers tires cornering force c3 80000Nmrad

the Centers of Gravity (CG) of the leading and trailing units, respectively. The

actuator is used to produce torque to steer the wheels on the trailer axle. In

addition, the trailer forward speed should be the same as the car forward speed

as the assumption made previously.

5.5 Single-lane Change Testing Maneuver

To conduct the designated RWA analysis, an open-loop testing procedure with a

single lane change (SLC) maneuver is simulated. A single-cycle sine-wave steer-

ing input with adjustable frequency and amplitude represents the input for the

SLC testing maneuver [79]. In this thesis, the frequency and the amplitude are

maintained constant at 0.4hz and 2.6degrees , respectively. Fig. 5.2, shows the

steering input used to test the performance measures of the combination.

Chapter 5 64

Figure 5.2: A representation of an SLC steering input

5.6 Testing Methodology

In previous chapters, details have been discussed, which have led to the use

of GDE3 as the MOEA for tuning the LQR controller. Thus, the same algo-

rithm is used to tune the LQR-based ATS controller for the car-trailer combi-

nation. A SLC maneuver is simulated to obtain lateral acceleration of the car

and trailer at various speeds. This test is performed in three cases: baseline de-

sign (without control system), the LQR-based controller, and GDE3 optimized

LQR controller (GDE3 optimized control system). During the SLC maneuver,

the control system uses the actuator on the trailers axel to steer the wheels for

generating the required cornering force to stabilize the trailer. The optimiza-

tion algorithm reads the simulation results, for each speed, and changes the

Chapter 5 65

gain matrices of the LQR controller to improve performance. Thus, the gain-

scheduling scheme considering the variation of the vehicle forward speed can

be implemented. The fitness test performance measures for each solution are

the RWA and peak-lateral acceleration of the car. The algorithm minimizes the

value of peak-lateral acceleration of the car, while trying to maintain an RWA

of 1.0.

5.7 Results and Discussions

This section presents the critical speed analysis and the simulation results of

the car-trailer combination with and without the ATS system.

5.7.1 Critical Speed Analysis

As defined earlier, the critical speed is determined as the stability boundary of

the car-trailer combination. Before testing the ATS system, the critical speed

for the baseline model is investigated. For unbiased testing, the same steering

input, as seen in Figure 5.2, is used to obtain simulation results at every speed.

The forward speed of the combination is the variable in this procedure and is

increased with an increment of 10km/h. It is observed that the critical speed

for the baseline system (without the ATS control) is 79.2km/h. Figures 5.3 and

5.4, show the response of the leading and trailing units, respectively, at critical

speed without ATS controller.

From Figures 5.3 and 5.4, the lateral acceleration of the car is around the spec-

ified boundaries, which means the combination is stable up to this speed. Fur-

thermore, by looking at Figure 5.4, the lateral acceleration of the trailer is larger

Chapter 5 66

Figure 5.3: Lateral accelera-tion of the car at the criti-cal speed 79.2km/h (without ATS

controller)

Figure 5.4: Lateral accelera-tion of the trailer at the criti-cal speed 79.2km/h (without ATS

controller)

in comparison to the lateral acceleration of the car. Thus, the trailer has an am-

plified lateral acceleration, which is also known as the RWA. The RWA of the

combination is calculated to be about 1.8. In other words, the trailers lateral

acceleration is 1.8 times greater than the cars lateral acceleration.

5.8 Forward Speed Variation

In this section, selected simulation results are compared, which are derived

from the baseline design (passive system without control), the LQR-based con-

troller, and the optimized LQR controller based on GDE3. Figs 5.5 to 5.8 show

the optimal Pareto-front for speeds from 80km/h to 110km/h. The x−axis shows

the error from the ideal RWA of 1 and the y − axis shows the peak lateral accel-

eration of the trailer.

Figs. 5.9 to 5.16 show the lateral acceleration of the car and trailer over the sim-

ulated SLC maneuver at the speed of 80km/h to 110km/h. The blue curve rep-

resents the passive system, the orange curve illustrates the ATS equipped with

Chapter 5 67

Figure 5.5: Optimal Pareto-frontat 80km/h




the LQR-based controller, and the black curve shows the response of the opti-

mized LQR controller. Compared with the passive system, the models with ATS

systems obviously suppress the peak values of the lateral accelerations. It can

be found that the lateral acceleration of the trailer is amplified in the passive

system when compared to the cars lateral acceleration. However, by looking at

the trailer equipped with the LQR and the optimized LQR controllers, it can be

seen that the lateral acceleration is significantly suppressed as anticipated.

From Figs. 5.9 to 5.16 it is observed that the LQR-based controller over-suppresses

the lateral acceleration of the trailer under the SLC maneuver. The over-suppressed

lateral acceleration of the trailer will degrade the path-following performance

Chapter 5 68

Figure 5.9: Lateral acceleration ofthe car at 80km/h

Figure 5.10: Lateral accelerationof the trailer at 80km/h

Figure 5.11: Lateral accelerationof the car at 90km/h




Chapter 5 69



Table 5.2: RWA Result Comparison.†

Speed (km/h) Passive LQR GDE3 LQR80 1.7959 0.3435 1.000190 1.8988 0.2857 1.0000

100 1.9745 0.2462 1.0000110 2.0337 0.2163 1.0000

† Ideal value of RWA is 1. RWA is calculated by applyingEq.(5.1) to Figs. 5.9 to 5.16.

Table 5.3: Peak Lateral Acceleration (g) of Trailer Result Comparison

Speed (km/h) Passive (g) LQR (g) GDE3 LQR (g) % Improvement†

80 0.3271 0.0481 0.1269 61.2090 0.4068 0.0440 0.1670 58.95

100 0.4823 0.0412 0.1742 63.88110 0.5575 0.0391 0.2065 62.96

† The % improvement is calculated w.r.t the passive system. Peak lateral accelerationshould be below 0.3g . %Improvement =| Vnew−Vold

Vold| ∗100, where V is the value of lateral

accelration.

of the combination, resulting in a less lateral displacement than the expected

under the SLC maneuver. In contrast to the case of the LQR-based controller,

the optimized LQR controller manipulates the lateral acceleration of the trailer,

and the RWA is effectively controlled around the value of 1.0, at which the lat-

eral stability and the path-following performance of the combination can be

improved simultaneously.

Chapter 5 70

Table 5.2 lists the RWA values of the system for speeds from 80km/h to 110km/h

to illustrate a comparison and show the percentage improvement achieved after

using a tuned controller. Table 5.3 shows the peak lateral acceleration of the

trailer at different speeds. These two measures are the objectives, which the

algorithm is tuning and are values of interest. From the aforementioned two

tables, it is clear that GDE3 tuned ATS controller performs by the far the best.

It not only avoids the pitfalls of over suppression, but ensures a more stable

system as well. These results are consistent with the results published in [79]

The above simulation results demonstrate that the optimized LQR controller

can well control the lateral motions of the leading and trailing vehicle units to

achieve desired and robust directional performance under the obstacle avoid-

ance maneuver at varied forward speeds. The supervisor performance of the

optimized LQR controller is attributed to the application of the gain schedul-

ing scheme and the optimization algorithm, GDE3.

5.9 Summary

Comparing the numerical results shown in Tables 5.2 and 5.3 reveals the fol-

lowing facts: (1) for the baseline design, the RWA ratio decreases with the ve-

hicle forward speed, which is consistent with the published results [79]; (2) the

LQR-based controller over-suppresses the lateral accelerations of both the lead-

ing and trailing vehicle units, leading to an excessive small RAW ratio and poor

path-following performance; (3) the optimized LQR controller can well con-

trol the lateral motions of both the leading and trailing vehicle units, achieve a

robust RWA ratio of 1.0.

Chapter 5 71

This chapter presents an innovative ATS controller, designed using the LQR

technique, the GDE3 optimization algorithm, and the first step towards a gain

scheduling scheme. Numerical simulations are conducted to test and validate

the effectiveness of the proposed robust ATS controller in terms of the direc-

tional performance of a car-trailer combination under a single lane change

maneuver at varied forward speeds. Simulation results demonstrate that the

proposed ATS controller can well control the lateral motions of both the lead-

ing and trailing vehicle units, effectively manipulate the RWA ratio around the

value of 1.0, and eventually achieve robust and improved lateral stability and

path-following performance. In Chapter 6 this concept is taken further to de-

velop a new gain scheduling controller.

Chapter 6

Design Optimization of Gain

Scheduling Controllers Using GDE3

and Closed-loop Co-Simulation

For a dynamic system model, a state-space representation is required to de-

sign an LQR-based controller. In the design of the LQR-based controller, the

weighting matrices, Q and R, should be determined in order to get the optimal

control gain matrix, K . In Chapter 5, a LQR-based ATS controller is optimized

using GDE3. In this chapter, a gain scheduling ATS controller is designed us-

ing GDE3 and closed-loop co-simulation. This method is proposed to generate

a two-dimensional lookup table, using driver model reaction time and vehicle

forward speed, for a gain-scheduled ATS system.

72

Chapter 6 73

6.1 Introduction

Car-Trailer combinations exhibit highly non-linear dynamics. In Chapter 5, the

car-trailer combination is linearized based on certain assumptions [72], which

leads to a 3-DOF car-trailer model. This model is then utilized to design and

evaluate an ATS controller. The open-loop dynamic simulation is carried out

using MATLAB/Simulink without a driver model. Due to the absence of a

driver model, a closed-loop SLC maneuver with a given trajectory cant be con-

ducted. Secondly, the 3-DOF linear model neglects some important dynamics

of the real-world car-trailer combinations, e.g., the lateral load transformation.

Thus, the fidelity and accuracy of the 3-DOF yaw-plane model may not be de-

sired. To address the above concerns, a new gain scheduling ATS controller is

designed using GDE3 and closed-loop co-simulation.

This chapter is an important starting point towards optimization using co-

simulation by combining the nonlinear car-trailer model developed in CarSim

and the controller designed in Simulink/Matlab. There is one system limi-

tations in using he CarSim model for the design optimization. The CarSim

Model can at most cater to 1500 generations by the optimizer. The simulation,

in contrast, may run for 100,000 generations. This greatly reduces the time and

opportunity for the algorithm to search the solution space. This limitation is

most probably a system specific issue, and in no way is a general problem with

CarSim software. Nevertheless, this limitation imposes the constraint for the

maximum generations to be achieved.

Chapter 6 74

Figure 6.1: Car sub-model developed in CarSim

Table 6.1: Vehicle parameter values.

Parameters ValueSprung Mass Car 1653kgRoll Inertia of Car 2765kgm2

Sprung Mass of Trailer 466kgRoll Inertia of Trailer 1810kg

6.2 CarSim Model and Methodology

CarSim is a mechanical simulation tool and is used to simulate automotive sys-

tems and analyze vehicle behavior. Based on analysis active controllers can be

developed and their performance tested and validated. It is a great tool for

active safety implementation and analysis. In this thesis, CarSim is used in

conjunction with MATLAB/Simulink to develop an ATS system. CarSim acts

as Software-in-Loop (SIL) and provides vehicle information required to tune

the controller.

The car-trailer model for the co-simulation is directly developed and tested in

CarSim software. Figs. 6.1 and 6.2 show the details of the car and trailer sub-

models used in the experiment. All the values are in meters. Table 6.1 lists the

car and trailer parameter values.

Fig. 6.3 shows the car-trailer model with the built-in driver model.

Chapter 6 75

Figure 6.2: Trailer sub-model developed in CarSim

Figure 6.3: Car-trailer model with the built-in driver model.

The co-simulation with CarSim model only requires the K matrix. One impor-

tant objective to bring this experiment to fruition is to greatly reduce the search

space so that the optimizer is able to find an optimal Pareto-front within 1500

generations. In order to achieve this, the K is generated directly with multi-

objective evolutionary algorithms.

6.2.1 Built-in Driver Model Offered from CarSim

In all previous tests of the car-trailer combination, open-loop simulations are

conducted with a given steering input. A single cycle sine-wave input is used

Chapter 6 76

Figure 6.4: Predefined trajectory for the closed-loop single lane-change testingmaneuver

to simulate a single lance-change maneuver. With a given steering input, de-

pending on the control gain, the trajectory of the car-trailer changes. In order to

ensure road safety, a proper maneuver (trajectory) must be ensured. In real life

driving, regardless of the speed or driver reaction time, a defined single lane-

change requires a car to move a fixed distance on the same road. To simulate a

defined single lane-change maneuver, the built-in driver model offered in Car-

Sim software is used. Fig. 6.4 shows the specified trajectory of the closed-loop

single lane-change maneuver. Over the testing maneuver, the driver model will

adaptively adjust its steering input in order to drive the vehicle to follow the

predefined trajectory. Two separate parameters of the driver model are used to

generate the look-up table for the GSC.

Chapter 6 77

Two separate parameters of the driver model are used to generate the look-up

table for the GSC. The two, driver model dependent, parameters are drivers

reaction time and vehicle forward speed.

In order to test the hypothesis, the co-simulation is run and it is found that

the design optimization using the MOEA is able to achieve an optimal Pareto-

front. The focal point of this section is a look-up table for the gain scheduling

controller (GSC) generated using GDE3.

6.2.2 Evaluation Criteria

Evaluation criteria used in this study is RWA and PFOT. Calculation of RWA

is very straightforward, the graphs of lateral acceleration of both the car and

trailer are acquired in Simulink. The peak lateral acceleration is calculated

from these graphs and used to calculate the RWA value.

PFOT is discussed in Chapter 5.3. There are many ways to measure PFOT [23].

The method to measure the PFOT used in this study is the peak overshoot from

the ideal trajectory of the car-trailer combination. The CarSim model generates

x and y coordinates of trajectory over the simulation period as well as the ideal

trajectory. A peak deviation from the trajectory (at any point) is calculated

from this graph and used as fitness criteria. To sum it up, the two fitness values

generated by the evaluation module are PFOT and RWA.

6.3 Gain Scheduling Controller

A gain scheduling controller (GSC) is used when a non-linear system can be

broken down into various linear operating ranges [80]. Control gain values,

Chapter 6 78

K are determined for each of these operating ranges to create a lookup ta-

ble. Based on environmental factors or internal operating conditions, the GSC

chooses a set of values from the look-up table. This allows for a more robust

non-linear control using linear techniques and gains scheduling [80]. There are

many variations of GSC based on how the parameters are varied [81], this the-

sis focuses on varying parameters based on vehicle forward speed and driver

reaction time to generate a two-dimensional lookup table. Ref. [81] provides a

survey into the success of gain scheduling controllers for non-linear control.

There are four steps to design and implement a GSC [81]. The first step is to

breakdown the existing system into a number of linearized models, as needed.

According to [81], a popular method to do this is the Jacobian linearization.

However, the approach used in this thesis does not require mathematical lin-

earization. By varying the speed and reaction time parameters in the CarSim

model, the model is automatically updated, which is utilized to generate op-

timal control gain parameters K for that particular scenario. The second step

is to design a linear controller [81], this is also covered by MOEA. The MOEA

driven ATS controller is the linear control method. This controller follows de-

sign constraints, natural selection, and biological evolution to generate control

gain K . The third step is to develop a look-up table and a look-up scheme [81].

This is the actual creation of the GSC. Each gain is scheduled based on vehicle

forward speed and reaction time of the driver model. After the GDE3 optimizer

terminates, an optimal Pareto-front is achieved. For each combination of vehi-

cle forward speed and driver reaction time, there exists an optimal Pareto-front.

To simplify the design, the best tradeoff value of each optimal Pareto-front is

used to test the GSC. The last step is to evaluate the systems performance [81]

Chapter 6 79

Table 6.2: Possibilities of GSC Look-up Table

Number Forward Speed (km/h) Reaction Time (s)1 80 02 90 03 100 04 110 05 120 06 80 0.17 90 0.18 100 0.19 110 0.1

10 120 0.1

and ensure that it is working within design parameters. The GDE3 optimized

controller and a gain scheduling controller are compared to confirm correct

functionality and to analyze the advantages.

The GSC works based on a two-dimensional look-up table. The first indepen-

dent parameter is the variation of vehicle forward speed, and the second inde-

pendent parameter is the reaction time of the driver model. Table 6.2 shows

all possible combinations. Each time a control gain value K is chosen for the

system, the forward speed of the vehicle and driver reaction time will be the

deciding parameters. Or in other words, based on the vehicle forward speed

and driver model’s reaction time the controller gain will be scheduled.

6.4 Modular Design Methodology

The term co-simulation, in this thesis, addresses the combined usage of CarSim

with MATLAB/Simulink. CarSim software offers an integrated S-Function for

the Simulink software package. The S-function data are sent and received by

CarSim. The simulation results are directly from the CarSim model and the

Chapter 6 80

Figure 6.5: System model using modular blocks. State Information from topto bottom: Yaw-rate of car, Yaw-rate of trailer, lateral speed of car and lateral

speed of trailer.

built-in testing maneuvers. This eliminates the chance of modeling errors and

ensures the soundness of the results.

The system model shown in Fig. 6.5 is highly modular. An important part

of this research is to ensure a method that is not system-specific. The system

model is based on 5 main blocks. The CarSim block holds the car-trailer model,

driver-model, and all built-in testing maneuvers. It also provides the simu-

lation results, which are fed into the system. The evaluation block receives

the simulation results from CarSim and assigns fitness values based on fitness

parameters. If the fitness values exceed or violate the constraints, they are ex-

cluded from the simulation. The optimization module receives the fitness costs

and uses them to assign fitness to population members for carrying out the evo-

lutionary algorithm and generating new parameters. The control module uses

the parameters from the optimization module to generate control gain matrix

K which is then fed as the ATS system’s steering angle to CarSim. Each of

Chapter 6 81

these blocks can be changed. The evaluation method, constraints, optimization

techniques, control strategies, and car-trailer model can all be tailored to any

system.

The control strategy, in the control and optimization modules, is evolutionary,

e.g., GDE3, NSGA-II, etc. This thesis proves this approach with five varying

evolutionary strategies, which will be discussed in section 6.6. As it works with

five variants, it can be safely said this approach should be successful with other

evolutionary algorithms.

The CarSim model changes parameter values every time a new speed or driver

model reaction time is introduced. This validates the modularity of the Car-

Sim block. Although not tested, it should work for software-in-loop simulation

software, which can provide state information.

The constraints module ensures that, above all, the control gain value must be

able to stabilize the system during the complete maneuver. The car-trailer com-

bination must follow the predefined path, without violating road boundaries.

The maximum allowed RWA of the car-trailer combination is 2.0. For the GSC,

GDE3 is used in the optimization module whereas the MOEA driven ATS con-

troller is hosted by the control module. The GDE3 will run for 1500 generations

with a population size of 48.

6.5 Overcoming System Limitation with Innoviza-

tion

An important part of what makes these experiments successful is the study of

the solutions in the Pareto-optimal front. In single objective optimization, there

Chapter 6 82

is a lack of data to formulate an opinion about the pattern that the solutions

follow. It is a common problem for optimization algorithms that they can get

stuck on local optima rather than achieve the global optima. A solution to this

problem is to use the results from MOEAs and reform the opinion about the

design choices [82]. In this thesis, the parameters evaluated are the search space

(SS) and the range of the design variables (DV). Due to the system limitation of

just 1000 iterations, innovization is extremely important to reach an optimal

Pareto-front with fewer iterations. If the SS is small and the variation of DV

is condensed, the algorithm automatically converges faster to a better solution.

Innovization is detailed in [82]. Instead of using an automated technique, this

task is carried out manually as per the steps listed in [82], for each speed and

reaction time pair.

There are two steps for using the innovization technique. First is the initial

declaration of SS and range of DV. At the start of the experiment, there is no

clear range of either SS or DV. The experiment is run and both parameters are

defined based on the maximum and minimum values in the Pareto-optimal

front. All algorithms listed in Fig. 6.6, apart from GDE3 Tuned, achieve their

optimal Pareto-front following the first step in establishing a baseline SS and

DV. For GDE3-Tuned, an additional innovization step was taken. This time, SS,

DV, and the rejection range or the constraints are reconsidered. SS and DV are

reinitialized based on new information and the constraints are changed from

design parameters to the range seen in the optimal Pareto-front shown in Fig.

6.6. By doing this, a better optimal Pareto-front is achieved.

Chapter 6 83

6.6 Algorithm Comparison

For the MOEA driven ATS controller, an algorithm must be decided. GDE3 is

a great algorithm for ATS controller [72], but it is not compared against other

MOEAs.

The five state-of-the-art MOEAs [83] are GDE3, Multi-objective Particle Swarm

Optimization (MOPSO) [84], Non-dominated Sorting Genetic Algorithm II [9]

and III [85] (NSGA II & III) and Strength Pareto Evolutionary Algorithm II

(SPEA2) [86]. The details of these algorithms are discussed in the referenced

articles. SPEA2, NSGA-II, and NSGA-III and GDE3 are rather similar, with

the main difference being their method to generate the optimal Pareto-front.

MOPSO, on the other hand, uses a completely different evolutionary technique

of Particle Swarm Optimization (PSO). The last algorithm is innovized GDE3.

By studying the optimal Pareto-front, the extreme points, and the knee-points,

the algorithm can be improved or innovized to provide even better results [82].

This is extremely important in this case, as only 1500 generations could be used

with a population size of 48.

After running the GDE3 algorithm and generating optimal Pareto-front at 120km/h,

each point is carefully studied. Each point in the optimal Pareto-front corre-

sponds to a control gain matrix, which has four terms in this case. By studying

the four terms and their maximum and minimum values, the search space is

reduced, and the GDE3 algorithm is run again. This is termed as tuned GDE3.

This test is extremely expensive and required a long time, and was not con-

ducted for all other algorithms nor all speeds. This is only displayed on the

comparison curve in Fig. 6.7.

Chapter 6 84

Figure 6.6: Algorithm comparison based on two design PFOT.

From Fig. 6.6 the points close to the origin are better as these are optimized

points.

The best algorithm is the tuned GDE3. The other two algorithms competing for

the first spot are GDE3 and SPEA2. GDE3 has more points and a better diver-

sity than SPEA2. The best point of individual objectives generated by GDE3 is

superior to SPEA2. These results along with the results from Chapter 5 prove

that GDE3 is the better algorithm. The x − axis of Fig. 6.6 illustrates the error

from the ideal RWA of 1.0 and the y − axis is the maximum deviation from the

ideal value in meters. Due to the expensive nature of testing and time limi-

tation, this same testing cannot be carried out on other speeds. The highest

speed is chosen as it presents the greatest difficulty. Figs. 6.7 and 6.8 show the

trajectory the vehicle followed in the driving simulation. These trajectories are

Chapter 6 85

Figure 6.7: Trajectory of Car at120km/h for Algorithm Compari-

son

Figure 6.8: Trajectory of Trailer at120km/h for Algorithm Compari-

son

Figure 6.9: Lateral Accelerationof Car at 120km/h for Algorithm

Comparison

Figure 6.10: Lateral Accelerationof Trailer at 120km/h for Algo-

rithm Comparison

calculated by using the trade-off point, the central point of the optimal Pareto-

front, for each of the algorithms. Figs. 6.9 and 6.10 show the lateral acceleration

of car and trailer respectively for the same simulation.

6.7 Results: GDE3 Generated GSC vs Passive

The optimal Pareto-front, for each speed and reaction time, has multiple values.

With a population size of 48, there can be up to 48 distinct possibilities of

control gain matrix K per schedule. It is not wasted labor to compare all points

Chapter 6 86

Figure 6.11: Optimal Pareto-frontfor 80 km/h and 0(s) driver model

reaction time

Figure 6.12: Optimal Pareto-frontfor 90 km/h and 0(s) driver model

reaction time

Figure 6.13: Optimal Pareto-frontfor 100 km/h and 0(s) driver

model reaction time


model reaction time

with all other points. There are three main points of interest in every optimal

Pareto-front: the two utopia points (extreme points) and the optimal tradeoff or

the best compromise solution. The optimal tradeoff is a decision to be made by

the system designer, but in this case, the point, which has the best compromise

between the two objectives, is considered.

Figs. 6.11 to 6.20 show the optimal Pareto-fronts, generated by GDE3, after co-

simulation using the system model and constraints specified in section 6.3. The

optimal Pareto-fronts for all possible studied cases, according to Table 6.2, are

listed. From each of these optimal Pareto-fronts, for each schedule, two utopia

Chapter 6 87


model reaction time

Figure 6.16: Optimal Pareto-frontfor 80 km/h and 0.1(s) driver

model reaction time


model reaction time


model reaction time


model reaction time


model reaction time

Chapter 6 88

Figure 6.21: Trajectory of the carat 80 km/h and 0(s) driver model

reaction time.

Figure 6.22: Trajectory of thetrailer at 80 km/h and 0(s) driver

model reaction time.


reaction time.



points and one optimal compromise are chosen, and the trajectories of car and

trailer are compared with the passive system.

Figs. 6.21 to 6.40 show the trajectories of car and trailer for all ten possible

schedules. As the speed increases, the value of the optimization algorithm and

the MOEA driven ATS controller are more apparent. Above 100km/h, the pas-

sive system completely fails to complete the maneuver when the driver model

reaction time is 0 seconds, and fails at 90km/h when the driver model reaction

time is set to 0.1 seconds. The GDE3 optimized ATS controller is able to sta-

bilize the system and to complete the single lane change maneuver. From Fig

Chapter 6 89


reaction time.




reaction time.




reaction time.



Chapter 6 90

Figure 6.31: Trajectory of thecar at 80 km/h and 0.1(s) driver


Figure 6.32: Trajectory of thetrailer at 80 km/h and 0.1(s)

driver model reaction time.

Figure 6.33: Trajectory of thecar at 90 km/h and 0.1(s) driver




Figure 6.35: Trajectory of the carat 100 km/h and 0.1(s) driver




Chapter 6 91









6.21, it can be observed that even though the maneuver is completed the RWA

value is around 0.45 for the RWA optimized point. From the progression of the

optimal Pareto-front for higher speeds and higher reaction times, the system

limitation plays an integral part. With only 1500 generations, it cannot be said

for certain that the best point has been found, for the hardest settings. The

results show that the GDE3 optimized MOEA ATS controller is indeed able to

stabilize the system to a great extent.

Chapter 6 92

Figure 6.41: GSC Trajectory of thecar at 80 km/h and 0.0(s) driver


Figure 6.42: GSC Trajectory ofthe trailer at 80 km/h and 0.0(s)






6.8 Results: Gain Scheduling

The last part of this chapter is to demonstrate, using the aforementioned re-

sults, how the gain scheduling controller (GSC) further improves the ATS sys-

tem. GSC controller is compared to a GDE3 optimized ATS tuned at 100km/h

at 0 second of the driver model reaction time. The trajectories of car and trailer

are compared. The control gains K is chosen such that the trajectory is opti-

mized.

From Figs. 6.42 to 6.58 it is apparent that using a GSC has a great advantage

Chapter 6 93













Chapter 6 94













Chapter 6 95





Table 6.3: RWA Comparison GS vs W/O GS

Speed (km/h) Reaction Time (s) GS W/O GS Improvement (%)80 0.0 1.0751 1.1749 8.4990 0.0 1.0929 1.2662 13.69

110 0.0 1.2072 1.3120 7.99120 0.0 1.3626 3.3314 59.0380 0.1 1.2503 1.2770 2.0990 0.1 1.3853 1.9563 29.19

100 0.1 1.4168 3.2507 56.42110 0.1 1.5600 3.2304 51.71120 0.1 1.8018 3.3320 45.92

over using a regular controller. For lower speeds, the improvement is not as

apparent by looking at just the trajectory, but looking at Table 6.3 the improve-

ment is more apparent. From this comprehensive study into gain scheduling

and co-simulation, it is demonstrated that GDE3 optimized GSC greatly im-

proves the performance of ATS for the non-linear car-trailer combination with

varying speeds and driver reaction time. This study includes both forward

speed variation and driver model reaction time variation. The simulation can

be further tuned towards creating a GSC that caters to the real driver as well as

the vehicle forward speed.

The GSC has a two-dimensional discrete scheme so an important note is when

Chapter 6 96

to switch between different modes of operation. In this thesis, two switching

methods are tested. Firstly, controller gains are changed when the speed in-

creases beyond the next discrete step, e.g. controller gains shifts from those

associated with 80km/h to 90km/h if the speed goes to or beyond 90km/h. The

second scheme tested is changing gains by rounding to the nearest value, e.g.

mode shifts from 80km/h to 90km/h when the speed crosses 85km/h. The per-

formance in both cases is seen to be similar but when using the first scheme the

overall number of times the gains change is less than the second.

6.9 Important Observation

The modular method proposed in this chapter utilizes a control strategy not

used before. Various MOEAs are used to generate control gain matrix K . It is

found that the proposed control strategy operates for all five algorithms. Sim-

ilarly, the modular approach works with 10 different combinations of vehicle

forward speed and driver model reaction time, as shown in Table 6.3. During

the process of innovization, the evaluation and constraint module is constantly

changed. When a different MOEA is used, the optimization and control mod-

ule also change. This is a comprehensive proof that this strategy is modular.

It conducted co-simulation is based on CarSim software. CarSim provides var-

ious information to the system, e.g. lateral accelerations, trajectories, etc. If

another software-in-loop (SIL) system or a hardware-in-loop (HIL) mechanical

simulator can provide similar information, this approach should work for that

particular tool too.

The modular method in section 6.4 utilizes a control strategy not used before.

Chapter 6 97

Various MOEA are used to generate control gain matrix K . The big observation

here is that it works, not just with one algorithm but 5. Similarly the modu-

lar approach works with 10 different CarSim models, anytime speed or driver

model reaction time is changed the model changes. During the process of in-

novization the evaluation and constraint module is constantly changed. When a

different MOEA is used the optimization and control module also change. This

is a comprehensive proof that this strategy is modular at least for CarSim. Car-

Sim provides various information back to the system, e.g. lateral accelerations,

trajectories, etc. If another software-in-loop (SIL) system or a hardware-in-loop

(HIL) mechanical simulator can provide the same information, this approach

should work for that particular tool too.

Chapter 7

Conclusions

This chapter concludes the thesis and highlights the insightful findings achieved

in the research. The motivation behind this thesis is to improve the lateral

stability of car-trailer combinations by designing robust ATS controllers using

multi-objective optimization. The knowledge retrieved from the results derived

from Chapters 3 and 4, as well as the detailed literature review conducted in

Chapter 2 is applied to the design optimization of the robust ATS controllers

implemented in Chapters 5 and 6.

7.1 Contributions

By means of a case study, an effective differential evolution mutation strategy

is tested and selected through a comprehensive performance and speed anal-

ysis. This strategy is used in the GDE3 variant for the design optimization of

the robust ATS controllers. Results show that this variant outperforms its ge-

netic algorithm counterpart, NSGA-II, and conventional tuning methods. Us-

ing GDE3, as the MOEA, an optimized ATS controller is obtained for car-trailer

98

Chapter 7 99

combinations. The system designer is able to choose a result from an optimal

Pareto-front set of solutions.

A MOEA tuned gain scheduling controller (GSC) is designed for car-trailer

combinations. The GSC is designed using the LQR control technique consid-

ering the variation of vehicle forward speed. A set of control gain matrices

are used as the lookup tables for the GSC. The GSC outperforms non-GSCs in

terms of improving the lateral stability and the PFOT performance of car-trailer

combinations.

Considering the interactions of driver-vehicle/controller-road, a new GSC is de-

signed using closed-loop co-simulation and multi-objective optimization method.

Two independent parameters are selected to build the two dimensional lookup

table for the gain scheduling scheme of the GSC. The two parameters are: 1)

driver model reaction time, and 2) vehicle forward speed. The co-simulation is

implemented by combining the controller designed in Simulink/Matlab with

the virtual car-trailer developed in CarSim. Built upon the co-simulation, a

multi-objective optimization method is applied to design the GSC. Numerical

simulation demonstrates the robustness and effectiveness of the GSC.

7.2 Recommendations for Future Studies

Based on the investigations conducted in the research, the following recom-

mendations are offered for future studies:

• The GSCs designed in this research are mainly focused on improving the

Chapter 7 100

lateral stability of car-trailer combinations under high-speed evasive ma-

neuvers. In the future studies, attention may be paid to extend the func-

tionality of the GSC for improving the low-speed PFOT in curved path

negotiations.

• The GDE3 Tuned algorithm performs well thanks to innovization. A fur-

ther study can be carried out on. Innovization is proven here to take the

existing algorithm to a greater place. By tackling both system limitations

and performing innovization, a more robust gain scheduling controller

may be created.

• The maneuvers simulated in this thesis are closed-loop single lane-change

maneuvers at a constant vehicle forward speed. A new testing maneu-

ver may be designed, which incorporates variable vehicle forward speeds.

This maneuver, with a greater speed sensitivity, may be used to further

improve and test the GSC. The research into the maneuver, which may

accurately depict and test a GSC, is a great step towards creating a robust

control system.

Bibliography

[1] Peter J Fleming and Robin C Purshouse. Evolutionary algorithms in con-

trol systems engineering: a survey. Control engineering practice, 10(11):

1223–1241, 2002.

[2] Aidan O’Dwyer. Handbook of PI and PID Controller Tuning Rules. Imperial

College Press, 2009. ISBN 9781848162426. URL https://books.google.

ca/books?id=zivrLUbIMgUC.

[3] Paul Fancher and Chris Winkler. Directional performance issues in eval-

uation and design of articulated heavy vehicles. Vehicle System Dynamics,

45, 07 2007.

[4] Der-Ho Wu. A theoretical study of the yaw/roll motions of a multiple

steering articulated vehicle. Proceedings of the Institution of Mechanical

Engineers, Part D: Journal of Automobile Engineering, 215(12):1257–1265,

2001.

[5] Rafay Shamim, Md Manjurul Islam, and Yuping He. A comparative study

of active control strategies for improving lateral stability of car-trailer sys-

tems. SAE Technical Papers, 04 2011.

[6] Gwen Moritz. The amazon effect. Arkansas Business, 34(26):18, Jun

2017. URL http://search.proquest.com.uproxy.library.dc-uoit.

ca/docview/1915756926?accountid=14694. Name - Amazon.com Inc;

Walmart Inc; Apple Inc; Copyright - Copyright Arkansas Business Jun 26-

Jul 2, 2017; Last updated - 2018-11-01.

[7] CTV Windsor. Transport trucks involved in 1 in 5 crashes

on ontario roads, 2017. URL https://windsor.ctvnews.ca/

transport-trucks-involved-in-1-in-5-crashes-on-ontario-roads-1.

3462207. [Online; accessed 2-January-2019].

101

https://books.google.ca/books?id=zivrLUbIMgUC

https://books.google.ca/books?id=zivrLUbIMgUC

http://search.proquest.com.uproxy.library.dc-uoit.ca/docview/1915756926?accountid=14694


https://windsor.ctvnews.ca/transport-trucks-involved-in-1-in-5-crashes-on-ontario-roads-1.3462207



Bibliography 102

[8] Hongliang Wang, Haobin Dong, Lianghua He, Yongle Shi, and Yuan

Zhang. Design and simulation of LQR controller with the linear inverted

pendulum. Electrical and Control Engineering (ICECE), 2010 International

Conference on, pages 699–702, 2010.

[9] Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, and TAMT Meyarivan. A

fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transac-

tions on evolutionary computation, 6(2):182–197, 2002.

[10] Manjurul Md Islam, Steve Mikaric, Yuping He, and Thomas Hu. Parallel

design optimization of articulated heavy vehicles with active safety sys-

tems. In Proceedings of the FISITA 2012 world automotive congress, pages

1563–1575. Springer, 2013.

[11] Yuping He, Md Manjurul Islam, and Timothy D Webster. An integrated

design method for articulated heavy vehicles with active trailer steering

systems. SAE International Journal of Passenger Cars-Mechanical Systems, 3

(2010-01-0092):158–174, 2010.

[12] Robert Dimeo and Kwang Y Lee. Boiler-turbine control system design

using a genetic algorithm. IEEE transactions on energy conversion, 10(4):

752–759, 1995.

[13] Nabil Nassif, Stanislaw Kajl, and Robert Sabourin. Optimization of hvac

control system strategy using two-objective genetic algorithm. HVAC&R

Research, 11(3):459–486, 2005.

[14] Jonathan A Wright, Heather A Loosemore, and Raziyeh Farmani. Opti-

mization of building thermal design and control by multi-criterion genetic

algorithm. Energy and buildings, 34(9):959–972, 2002.

[15] Kailash Krishnaswamy, George Papageorgiou, Sonja Glavaski, and Anto-

nis Papachristodoulou. Analysis of aircraft pitch axis stability augmenta-

tion system using sum of squares optimization. In Proceedings of the 2005,

American Control Conference, 2005., pages 1497–1502. IEEE, 2005.

[16] Mahesh P Nagarkar and Gahininath Vikhe. Optimization of the linear

quadratic regulator (LQR) control quarter car suspension system using ge-

netic algorithm. Ingenierıa e Investigacion, 36(1):23–30, 2016.

Bibliography 103

[17] Amir Ghoreishi and Mohammad Nekoui. Optimal weighting matrices de-

sign for LQR controller based on genetic algorithm and pso. Advanced

Materials Research, 433-440:7546–7553, 01 2012.

[18] Yuping He and Md Manjurul Islam. An automated design method for

active trailer steering systems of articulated heavy vehicles. Journal of Me-

chanical Design, 134(4):041002, 2012.

[19] Eungkil Lee, Saurabh Kapoor, Tushita Sikder, and Yuping He. An opti-

mal robust controller for active trailer differential braking systems of car-

trailer combinations. International Journal of Vehicle Systems Modelling and

Testing, 12(1-2):72–93, 2017.

[20] Rudolf Emil Kalman. Contributions to the theory of optimal control. Bol.

Soc. Mat. Mexicana, 5(2):102–119, 1960.

[21] Rudolph Emil Kalman. A new approach to linear filtering and prediction

problems. Journal of basic Engineering, 82(1):35–45, 1960.

[22] Amin Mohammadbagheri, Narges Zaeri, and Mahdi Yaghoobi. Compar-

ison performance between PID and LQR controllers for 4-leg voltage-

source inverters. In International Conference Circuit, System and Simulation,

2011.

[23] Md. Manjurul Islam. Design synthesis of articulated heavy vehicles with

active trailer steering systems. Master’s thesis, University of Ontario In-

stitute of Technology, Oshawa, Ontario, Canada, 2010.

[24] Van Tan Vu, Olivier Sename, Luc Dugard, and Pter Gspr. Active anti-roll

bar control using electronic servo valve hydraulic damper on single unit

heavy vehicle. IFAC-PapersOnLine, 49:418–425, 12 2016. doi: 10.1016/j.

ifacol.2016.08.062.

[25] Yuping He and Md Manjurul Islam. An automated design method for

active trailer steering systems of articulated heavy vehicles. Journal of Me-

chanical Design, 134(4):041002, 2012.

[26] Kyong-il Kim, Hsin Guan, Bo Wang, Rui Guo, and Fan Liang. Active steer-

ing control strategy for articulated vehicles. Frontiers of Information Tech-

nology & Electronic Engineering, 17(6):576–586, Jun 2016.

Bibliography 104

[27] Guido Koch and Tobias Kloiber. Driving state adaptive control of an active

vehicle suspension system. IEEE transactions on Control Systems technology,

22(1):44–57, 2013.

[28] Xuejun Ding, Yuping He, Jing Ren, and Tao Sun. A comparative study of

control algorithms for active trailer steering systems of articulated heavy

vehicles. In 2012 American Control Conference (ACC), pages 3617–3622.

IEEE, 2012.

[29] Arjun C Unni, Anjali Junghare, Vivek Mohan, and Weerakorn Ongsakul.

PID, fuzzy and LQR controllers for magnetic levitation system. In 2016

International Conference on Cogeneration, Small Power Plants and District

Energy (ICUE), pages 1–5. IEEE, 2016.

[30] Mundher H.A. Yaseen. A comparative study of stabilizing control of a

planer electromagnetic levitation using PID and LQR controllers. Results

in Physics, 7:4379 – 4387, 2017. ISSN 2211-3797.

[31] Herman A Reise. Automatic trailer sway sensing and brake applying sys-

tem, August 9 1977. US Patent 4,040,507.

[32] Mutaz Keldani and Yuping He. Design of electronic stability control (esc)

systems for car-trailer combinations. EasyChair Preprint no. 313, Easy-

Chair, 2018. doi: 10.29007/hrw2.

[33] Ning Zhang, Guo-dong Yin, Tian Mi, Xiao-gao Li, and Nan Chen. Analysis

of dynamic stability of car-trailer combinations with nonlinear damper

properties. Procedia Iutam, 22:251–258, 2017.

[34] John T Kasselmann, Henry Dorsett, and Harold J Burkett. Sway control

means for a trailer, April 6 1976. US Patent 3,948,567.

[35] Paul Fancher and Chris Winkler. Directional performance issues in eval-

uation and design of articulated heavy vehicles. Vehicle System Dynamics,

45, 07 2007.

[36] Yuping He. Design of Rail Vehicles with Passive and Active Suspensions Us-

ing Multidisciplinary Optimization, Multibody Dynamics, and Genetic Algo-

rithms. PhD thesis, University of Waterloo, Waterloo, Ontario, Canada,

2003.

Bibliography 105

[37] Tao Sun, Yuping He, Ebrahim Esmailzadeh, and Jing Ren. Lateral stability

improvement of car-trailer systems using active trailer braking control.

Journal of Mechanics Engineering and Automation, 2(9):555–562, 2012.

[38] Ossama Mokhiamar. Stabilization of car-caravan combination using inde-

pendent steer and drive/or brake forces distribution. Alexandria Engineer-

ing Journal, 54(3):315–324, 2015.

[39] Seshadri Sankar, Seshardri Sankar, Subhash Rakheja, and Alain Piche. Di-

rectional dynamics of a tractor-semitrailer with self-and forced-steering

axles. SAE transactions, pages 528–541, 1991.

[40] Yuping He, Hoda Elmaraghy, and Waguih Elmaraghy. A design analysis

approach for improving the stability of dynamic systems with application

to the design of car-trailer systems. Modal Analysis, 11(12):1487–1509,

2005.

[41] Andrew MC Odhams, Richard L Roebuck, David Cebon, and Chris Win-

kler. Dynamic safety of active trailer steering systems. Proceedings of the

Institution of Mechanical Engineers, Part K: Journal of multi-body dynamics,

222(4):367–380, 2008.

[42] Krishna Rangavajhula and HS Jacob Tsao. Command steering of trailers

and command-steering-based optimal control of an articulated system for

tractor-track following. Proceedings of the Institution of Mechanical Engi-

neers, Part D: Journal of Automobile Engineering, 222(6):935–954, 2008.

[43] Moustafa Ei-Gindy, Nezih Mrad, and Xiaoya Tong. Sensitivity of rearward

amplication control of a truck/full trailer to tyre cornering stiffness varia-

tions. Proceedings of the Institution of Mechanical Engineers, Part D: Journal

of Automobile Engineering, 215(5):579–588, 2001.

[44] Paul Fancher, Chris Winkler, Robert Ervin, and Hongli Zhang. Using brak-

ing to control the lateral motions of full trailers. Vehicle System Dynamics,

29(S1):462–478, 1998.

[45] Darrell Etherington. The tesla semi tackles this classic truck safety

problem using tech,, 2017. URL https://techcrunch.com/2017/11/16/

the-tesla-semi-tackles-this-classic-truck-safety-problem-using-tech/.

https://techcrunch.com/2017/11/16/the-tesla-semi-tackles-this-classic-truck-safety-problem-using-tech/

https://techcrunch.com/2017/11/16/the-tesla-semi-tackles-this-classic-truck-safety-problem-using-tech/

Bibliography 106

[46] Rainer Storn and Kenneth Price. Differential evolution–a simple and effi-

cient heuristic for global optimization over continuous spaces. Journal of

global optimization, 11(4):341–359, 1997.

[47] Gurusamy Jeyakumar and C. Shunmuga Velayutham. Distributed mixed

variant differential evolution algorithms for unconstrained global opti-

mization. Memetic Computing, 5, 12 2013.

[48] Muhammed Dangor, Olurotimi A Dahunsi, Jimoh O Pedro, and M Montaz

Ali. Evolutionary algorithm-based PID controller tuning for nonlinear

quarter-car electrohydraulic vehicle suspensions. Nonlinear dynamics, 78

(4):2795–2810, 2014.

[49] Edward M Kasprzak and Kemper Lewis. An approach to facilitate decision

tradeoffs in pareto solution sets. Journal of Engineering Valuation and Cost

Analysis, 3:173–187, 01 2000.

[50] Song Zhang, Hongfeng Wang, and Min Huang. Dominate gradient strat-

egy based on pareto dominant and gradient method. In 2016 Chinese Con-

trol and Decision Conference (CCDC), pages 4943–4948. IEEE, 2016.

[51] Kalyanmoy Deb and Shivam Gupta. Understanding knee points in bi-

criteria problems and their implications as preferred solution principles.

Engineering optimization, 43(11):1175–1204, 2011.

[52] Joshua Knowles and David Corne. The pareto archived evolution strategy:

A new baseline algorithm for pareto multiobjective optimisation. Proceed-

ings of the 1999 Congress on Evolutionary Computation, CEC 1999, 1, 01

1999.

[53] E. Zitzler. Evolutionary algorithms for multiobjective optimization: Methods

and applications. PhD thesis, Swiss Federal Institute of Technology, Zurich,

Switzerland, 1999.

[54] Varvara Mytilinou and Athanasios J. Kolios. A multi-objective optimisa-

tion approach applied to offshore wind farm location selection. Journal of

Ocean Engineering and Marine Energy, 3(3):265–284, 2017.

Bibliography 107

[55] Franklin Mendoza, Jose L Bernal-Agustin, and Jose A Domınguez-

Navarro. NSGA and SPEA applied to multiobjective design of power dis-

tribution systems. IEEE Transactions on power systems, 21(4):1938–1945,

2006.

[56] Arindam Deb, Jibendu Roy, and Bhaskar Gupta. Performance comparison

of differential evolution, particle swarm optimization and genetic algo-

rithm in the design of circularly polarized microstrip antennas. Anten-

nas and Propagation, IEEE Transactions on, 62:3920–3928, 08 2014. doi:

10.1109/TAP.2014.2322880.

[57] Saku Kukkonen and Jouni Lampinen. GDE3: The third evolution step of

generalized differential evolution. In 2005 IEEE congress on evolutionary

computation, volume 1, pages 443–450. IEEE, 2005.

[58] Thomas L. Magnanti. Twenty Years of Mathematical Programming. 1989.

URL https://books.google.ca/books?id=9xiaoAEACAAJ.

[59] Anagha Philip Antony and Elizabeth Varghese. Comparison of perfor-

mance indices of PID controller with different tuning methods. In 2016

International Conference on Circuit, Power and Computing Technologies (IC-

CPCT), pages 1–6. IEEE, 2016.

[60] Jussara MS Ribeiro, Murillo F Santos, MJ Carmo, and MF Silva. Compari-

son of PID controller tuning methods: analytical/classical techniques ver-

sus optimization algorithms. In 2017 18th international Carpathian control

conference (ICCC), pages 533–538. IEEE, 2017.

[61] Jakob Vesterstrom and Rene Thomsen. A comparative study of differen-

tial evolution, particle swarm optimization, and evolutionary algorithms

on numerical benchmark problems. In Proceedings of the 2004 Congress

on Evolutionary Computation (IEEE Cat. No. 04TH8753), volume 2, pages

1980–1987. IEEE, 2004.

[62] Nurhan Karaboga and Bahadir Cetinkaya. Performance comparison of ge-

netic and differential evolution algorithms for digital fir filter design. In

International Conference on Advances in Information Systems, pages 482–

488. Springer, 2004.

https://books.google.ca/books?id=9xiaoAEACAAJ

Bibliography 108

[63] Malcolm JA Strens, Mark Bernhardt, and Nicholas Everett. Markov chain

monte carlo sampling using direct search optimization. In ICML, pages

602–609. Citeseer, 2002.

[64] Manasa Madhavi Puralachetty and Vinay Kumar Pamula. Differential

evolution and particle swarm optimization algorithms with two stage ini-

tialization for PID controller tuning in coupled tank liquid level system.

In 2016 International Conference on Advanced Robotics and Mechatronics

(ICARM), pages 507–511. IEEE, 2016.

[65] Andri Mirzal, Shinichiro Yoshii, and Masashi Furukawa. PID parameters

optimization by using genetic algorithm. arXiv preprint arXiv:1204.0885,

2012.

[66] Michigan State University, Carnegie Mellon University, and University

of Detroit Mercy. Control tutorials for matlab simulink. URL http:

//ctms.engin.umich.edu/CTMS/index.php?aux=Home.

[67] Fateme Azimlu, Shahryar Rahnamayan, Masoud Makrehchi, and Pedram

Karimipour-Fard. Designing solar chimney power plant using meta-

modeling, multi-objective optimization, and innovization. In International

Conference on Evolutionary Multi-Criterion Optimization, pages 731–742.

Springer, 2019.

[68] Zakiya Alfughi, Shahryar Rahnamayan, and Bekir Yilbas. Multi-objective

solar farm design based on parabolic collectors. Journal of Advanced Com-

putational Intelligence and Intelligent Informatics, 22(2):256–270, 2018.

[69] Brian Jujnovich and David Cebon. Comparative performance of semi-

trailer steering systems. In 7th International Symposium on Heavy Vehicle

Weights & Dimensions, Delft, The Netherlands, Europe, 2002.

[70] Ashish Tewari. Advanced control of aircraft, spacecraft and rockets, vol-

ume 37. John Wiley & Sons, 2011.

[71] Ruina Dang, Jianqiang Wang, Shengbo Eben Li, and Keqiang Li. Coordi-

nated adaptive cruise control system with lane-change assistance. IEEE

Transactions on Intelligent Transportation Systems, 16(5):2373–2383, 2015.

[72] Mutaz Keldani, Khizar Qureshi, Yuping He, and Ramiro Liscano. Design

and optimization of a robust active trailer steering system for car-trailer

http://ctms.engin.umich.edu/CTMS/index.php?aux=Home

http://ctms.engin.umich.edu/CTMS/index.php?aux=Home

Bibliography 109

combinations. In SAE Technical Paper. SAE International, 04 2019. doi:

10.4271/2019-01-0433. URL https://doi.org/10.4271/2019-01-0433.

[73] Bang G. Zheng, Yi B. Huang, and Cong Y. Qiu. LQR+PID control and im-

plementation of two-wheeled self-balancing robot. Applied Mechanics and

Materials, 590:399–406, 06 2014. URL http://search.proquest.com.

uproxy.library.dc-uoit.ca/docview/1542940398?accountid=14694.

Copyright - Copyright Trans Tech Publications Ltd. Jun 2014; Last

updated - 2014-12-09.

[74] Robert C. Nelson. Flight stability and automatic control. McGraw-Hill series

in aeronautical and aerospace engineering. McGraw-Hill Ryerson, Lim-

ited, 1989. ISBN 9780070462182.

[75] Hans Prem, Austroads, and Australia. National Road Transport Commis-

sion. Comparison of Modelling Systems for Performance-based Assessments

of Heavy Vehicles: (performance Based Standards NRTC/Austroads Project

A3 and A4) : Working Paper. National Road Transport Commission,

2001. ISBN 9780642544896. URL https://books.google.ca/books?id=

XrN-YgEACAAJ.

[76] Yuping He, Amir Khajepour, J McPhee, and Xiaohui Wang. Dynamic mod-

elling and stability analysis of articulated frame steer vehicles. Interna-

tional Journal of Heavy Vehicle Systems, 12(1):28–59, 2004.

[77] B. P. Minaker and R. J. Rieveley. Automatic generation of the non-

holonomic equations of motion for vehicle stability analysis. Vehicle Sys-

tem Dynamics, 48(9):1043–1063, 2010. doi: 10.1080/00423110903248702.

URL https://doi.org/10.1080/00423110903248702.

[78] MFJ Luijten. Lateral dynamic behaviour of articulated commercial vehi-

cles. Eindhoven University of Technology, 2010.

[79] Qiushi Wang and Yuping He. A study on single lane-change manoeuvres

for determining rearward amplification of multi-trailer articulated heavy

vehicles with active trailer steering systems. Vehicle System Dynamics, 54

(1):102–123, 2016.

[80] Douglas A Lawrence and Wilson J Rugh. Gain scheduling dynamic linear

controllers for a nonlinear plant. Automatica, 31(3):381–390, 1995.

https://doi.org/10.4271/2019-01-0433



https://books.google.ca/books?id=XrN-YgEACAAJ

https://books.google.ca/books?id=XrN-YgEACAAJ

https://doi.org/10.1080/00423110903248702

Bibliography 110

[81] Wilson J Rugh and Jeff S Shamma. Research on gain scheduling. Automat-

ica, 36(10):1401–1425, 2000.

[82] Kalyanmoy Deb and Aravind Srinivasan. Innovization: Innovating design

principles through optimization. In Proceedings of the 8th annual conference

on Genetic and evolutionary computation, pages 1629–1636. ACM, 2006.

[83] Joshua B Kollat and Patrick M Reed. Comparing state-of-the-art evolu-

tionary multi-objective algorithms for long-term groundwater monitoring

design. Advances in Water Resources, 29(6):792–807, 2006.

[84] CA Coello Coello and M Salazar Lechuga. MOPSO: A proposal for mul-

tiple objective particle swarm optimization. In Proceedings of the 2002

Congress on Evolutionary Computation. CEC’02 (Cat. No. 02TH8600), vol-

ume 2, pages 1051–1056. IEEE, 2002.

[85] Kalyanmoy Deb and Himanshu Jain. An evolutionary many-objective op-

timization algorithm using reference-point-based nondominated sorting

approach, part i: solving problems with box constraints. IEEE transactions

on evolutionary computation, 18(4):577–601, 2013.

[86] Eckart Zitzler, Marco Laumanns, and Lothar Thiele. SPEA2: Improving

the strength pareto evolutionary algorithm. TIK-report, 103, 2001.

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